{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0001","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a8c60491b3224c4d38fa2da2fd87df049f47ecc537e0bec753374364dd3043d","family_id":"qf-sequence-limit-reciprocal-value","seed":202610081001,"semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"-12\"],\"operation\":\"reciprocal_sequence_limit\"}"},"fields":{"title":"Limit of an Explicit Reciprocal-Tail Sequence","stem":"On natural indices starting at zero, u(n)=(-12)+(0)/(n+1), with d≥1. Find the declared reciprocal-tail limit.","answer":{"kind":"short_text","value":"-12","explanation":"The reciprocal error tends to zero in this model, leaving L=-12, including c=0. This is not an inference about an arbitrary sequence from finitely many terms."},"skills":["Limit of an Explicit Reciprocal-Tail Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0002","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3d059a9c401500ff6d8036942109e8af91c690af54fa5bef0c080271f00b1fd","family_id":"qf-sequence-limit-reciprocal-value","seed":202610081002,"semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"-11\"],\"operation\":\"reciprocal_sequence_limit\"}"},"fields":{"title":"Limit of an Explicit Reciprocal-Tail Sequence","stem":"On natural indices starting at zero, u(n)=(-11)+(-2)/(n+2), with d≥1. Find the declared reciprocal-tail limit.","answer":{"kind":"short_text","value":"-11","explanation":"The reciprocal error tends to zero in this model, leaving L=-11, including c=0. This is not an inference about an arbitrary sequence from finitely many terms."},"skills":["Limit of an Explicit Reciprocal-Tail Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0003","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af45aeb70640dad44943b6e99e6b0261eb4c122f6d408151137c064ff1b0a520","family_id":"qf-sequence-limit-reciprocal-value","seed":202610081003,"semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"-10\"],\"operation\":\"reciprocal_sequence_limit\"}"},"fields":{"title":"Limit of an Explicit Reciprocal-Tail Sequence","stem":"On natural indices starting at zero, u(n)=(-10)+(3)/(n+3), with d≥1. Find the declared reciprocal-tail limit.","answer":{"kind":"short_text","value":"-10","explanation":"The reciprocal error tends to zero in this model, leaving L=-10, including c=0. This is not an inference about an arbitrary sequence from finitely many terms."},"skills":["Limit of an Explicit Reciprocal-Tail Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0004","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11f9a0d94cb0991386ca064ba2719e2093d57e4257dacd271c7f37bf392cf4d6","family_id":"qf-sequence-limit-reciprocal-value","seed":202610081004,"semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"-9\"],\"operation\":\"reciprocal_sequence_limit\"}"},"fields":{"title":"Limit of an Explicit Reciprocal-Tail Sequence","stem":"On natural indices starting at zero, u(n)=(-9)+(-4)/(n+4), with d≥1. Find the declared reciprocal-tail limit.","answer":{"kind":"short_text","value":"-9","explanation":"The reciprocal error tends to zero in this model, leaving L=-9, including c=0. This is not an inference about an arbitrary sequence from finitely many terms."},"skills":["Limit of an Explicit Reciprocal-Tail Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0005","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4943d24ae902ec7ec3f80b5a349cccdd5b07ab20137a459209ee6900cdc1d3d","family_id":"qf-sequence-limit-reciprocal-value","seed":202610081005,"semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"-8\"],\"operation\":\"reciprocal_sequence_limit\"}"},"fields":{"title":"Limit of an Explicit Reciprocal-Tail Sequence","stem":"On natural indices starting at zero, u(n)=(-8)+(5)/(n+5), with d≥1. Find the declared reciprocal-tail limit.","answer":{"kind":"short_text","value":"-8","explanation":"The reciprocal error tends to zero in this model, leaving L=-8, including c=0. This is not an inference about an arbitrary sequence from finitely many terms."},"skills":["Limit of an Explicit Reciprocal-Tail Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0006","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ceb38294187cb0ee6e872855653023983db5dedd924b036228bcd34d3a53303","family_id":"qf-sequence-limit-reciprocal-value","seed":202610081006,"semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"-7\"],\"operation\":\"reciprocal_sequence_limit\"}"},"fields":{"title":"Limit of an Explicit Reciprocal-Tail Sequence","stem":"On natural indices starting at zero, u(n)=(-7)+(-6)/(n+1), with d≥1. Find the declared reciprocal-tail limit.","answer":{"kind":"short_text","value":"-7","explanation":"The reciprocal error tends to zero in this model, leaving L=-7, including c=0. This is not an inference about an arbitrary sequence from finitely many terms."},"skills":["Limit of an Explicit Reciprocal-Tail Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0007","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a89f32b4fea257e3075a0796ef0af7e5b6c6fef4b88ffc6463b335f9e589661d","family_id":"qf-sequence-limit-reciprocal-value","seed":202610081007,"semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"-6\"],\"operation\":\"reciprocal_sequence_limit\"}"},"fields":{"title":"Limit of an Explicit Reciprocal-Tail Sequence","stem":"On natural indices starting at zero, u(n)=(-6)+(7)/(n+2), with d≥1. Find the declared reciprocal-tail limit.","answer":{"kind":"short_text","value":"-6","explanation":"The reciprocal error tends to zero in this model, leaving L=-6, including c=0. This is not an inference about an arbitrary sequence from finitely many terms."},"skills":["Limit of an Explicit Reciprocal-Tail Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0008","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"655e207480cdd020ce5c750a9b073ddd77981431ac0ff6b1b869ce270170329c","family_id":"qf-sequence-limit-reciprocal-value","seed":202610081008,"semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"-5\"],\"operation\":\"reciprocal_sequence_limit\"}"},"fields":{"title":"Limit of an Explicit Reciprocal-Tail Sequence","stem":"On natural indices starting at zero, u(n)=(-5)+(-8)/(n+3), with d≥1. Find the declared reciprocal-tail limit.","answer":{"kind":"short_text","value":"-5","explanation":"The reciprocal error tends to zero in this model, leaving L=-5, including c=0. This is not an inference about an arbitrary sequence from finitely many terms."},"skills":["Limit of an Explicit Reciprocal-Tail Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0009","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3bc137b2508a5b4626e0ba81690e23988fa32fa3d41eecceec24afaa681b08f6","family_id":"qf-sequence-limit-reciprocal-value","seed":202610081009,"semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"-4\"],\"operation\":\"reciprocal_sequence_limit\"}"},"fields":{"title":"Limit of an Explicit Reciprocal-Tail Sequence","stem":"On natural indices starting at zero, u(n)=(-4)+(9)/(n+4), with d≥1. Find the declared reciprocal-tail limit.","answer":{"kind":"short_text","value":"-4","explanation":"The reciprocal error tends to zero in this model, leaving L=-4, including c=0. This is not an inference about an arbitrary sequence from finitely many terms."},"skills":["Limit of an Explicit Reciprocal-Tail Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0010","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b0e1dc19930021f9787c35bfa4d115c7b160c9f37539609bd9850133d4fe858","family_id":"qf-sequence-limit-reciprocal-value","seed":202610081010,"semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"-3\"],\"operation\":\"reciprocal_sequence_limit\"}"},"fields":{"title":"Limit of an Explicit Reciprocal-Tail Sequence","stem":"On natural indices starting at zero, u(n)=(-3)+(-10)/(n+5), with d≥1. Find the declared reciprocal-tail limit.","answer":{"kind":"short_text","value":"-3","explanation":"The reciprocal error tends to zero in this model, leaving L=-3, including c=0. This is not an inference about an arbitrary sequence from finitely many terms."},"skills":["Limit of an Explicit Reciprocal-Tail Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0011","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49fb9fcfa9318884aa52c4b39ae640c10a4553824a49b214bfaf35c1ae446007","family_id":"qf-sequence-limit-reciprocal-value","seed":202610081011,"semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"-2\"],\"operation\":\"reciprocal_sequence_limit\"}"},"fields":{"title":"Limit of an Explicit Reciprocal-Tail Sequence","stem":"On natural indices starting at zero, u(n)=(-2)+(11)/(n+1), with d≥1. Find the declared reciprocal-tail limit.","answer":{"kind":"short_text","value":"-2","explanation":"The reciprocal error tends to zero in this model, leaving L=-2, including c=0. This is not an inference about an arbitrary sequence from finitely many terms."},"skills":["Limit of an Explicit Reciprocal-Tail Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0012","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a9a75a80c61f6085c2d98241015f5e459c028c6aa9fd94715f6ad6b43fa72b8","family_id":"qf-sequence-limit-reciprocal-value","seed":202610081012,"semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"-1\"],\"operation\":\"reciprocal_sequence_limit\"}"},"fields":{"title":"Limit of an Explicit Reciprocal-Tail Sequence","stem":"On natural indices starting at zero, u(n)=(-1)+(-12)/(n+2), with d≥1. Find the declared reciprocal-tail limit.","answer":{"kind":"short_text","value":"-1","explanation":"The reciprocal error tends to zero in this model, leaving L=-1, including c=0. This is not an inference about an arbitrary sequence from finitely many terms."},"skills":["Limit of an Explicit Reciprocal-Tail Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0013","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3225ac477f439f1ecffb35cf39ddb82693ebcde919612604f3a810c2ba3b1652","family_id":"qf-sequence-limit-reciprocal-value","seed":202610081013,"semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"0\"],\"operation\":\"reciprocal_sequence_limit\"}"},"fields":{"title":"Limit of an Explicit Reciprocal-Tail Sequence","stem":"On natural indices starting at zero, u(n)=(0)+(13)/(n+3), with d≥1. Find the declared reciprocal-tail limit.","answer":{"kind":"short_text","value":"0","explanation":"The reciprocal error tends to zero in this model, leaving L=0, including c=0. This is not an inference about an arbitrary sequence from finitely many terms."},"skills":["Limit of an Explicit Reciprocal-Tail Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0014","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e24bdd6ed636b47beea6c63454b92969a7c427635b2aa3c922d657697f49c1dc","family_id":"qf-sequence-limit-reciprocal-value","seed":202610081014,"semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"1\"],\"operation\":\"reciprocal_sequence_limit\"}"},"fields":{"title":"Limit of an Explicit Reciprocal-Tail Sequence","stem":"On natural indices starting at zero, u(n)=(1)+(-14)/(n+4), with d≥1. Find the declared reciprocal-tail limit.","answer":{"kind":"short_text","value":"1","explanation":"The reciprocal error tends to zero in this model, leaving L=1, including c=0. This is not an inference about an arbitrary sequence from finitely many terms."},"skills":["Limit of an Explicit Reciprocal-Tail Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0015","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f27421110588d213378ac883cadfad35bc1d4ecf119bee9a1ab4e3f74ff89bd2","family_id":"qf-sequence-limit-reciprocal-value","seed":202610081015,"semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"2\"],\"operation\":\"reciprocal_sequence_limit\"}"},"fields":{"title":"Limit of an Explicit Reciprocal-Tail Sequence","stem":"On natural indices starting at zero, u(n)=(2)+(15)/(n+5), with d≥1. Find the declared reciprocal-tail limit.","answer":{"kind":"short_text","value":"2","explanation":"The reciprocal error tends to zero in this model, leaving L=2, including c=0. This is not an inference about an arbitrary sequence from finitely many terms."},"skills":["Limit of an Explicit Reciprocal-Tail Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0016","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7545360f7fe54713786dc7fdcff7b702c5753f29fd5b860dea9e600b09707cfb","family_id":"qf-sequence-limit-reciprocal-value","seed":202610081016,"semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"3\"],\"operation\":\"reciprocal_sequence_limit\"}"},"fields":{"title":"Limit of an Explicit Reciprocal-Tail Sequence","stem":"On natural indices starting at zero, u(n)=(3)+(-16)/(n+1), with d≥1. Find the declared reciprocal-tail limit.","answer":{"kind":"short_text","value":"3","explanation":"The reciprocal error tends to zero in this model, leaving L=3, including c=0. This is not an inference about an arbitrary sequence from finitely many terms."},"skills":["Limit of an Explicit Reciprocal-Tail Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0017","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4135fae6d66817200cb2a03a3cd119802d2d6b1ce9c07bc2b6ea4092c4a6a2b","family_id":"qf-sequence-limit-reciprocal-value","seed":202610081017,"semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"4\"],\"operation\":\"reciprocal_sequence_limit\"}"},"fields":{"title":"Limit of an Explicit Reciprocal-Tail Sequence","stem":"On natural indices starting at zero, u(n)=(4)+(17)/(n+2), with d≥1. Find the declared reciprocal-tail limit.","answer":{"kind":"short_text","value":"4","explanation":"The reciprocal error tends to zero in this model, leaving L=4, including c=0. This is not an inference about an arbitrary sequence from finitely many terms."},"skills":["Limit of an Explicit Reciprocal-Tail Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0018","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f5f6b954110cf316dc60df3bb17b89df3be1fad2225c406cae3f4acfcf5d3e3","family_id":"qf-sequence-limit-reciprocal-value","seed":202610081018,"semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"5\"],\"operation\":\"reciprocal_sequence_limit\"}"},"fields":{"title":"Limit of an Explicit Reciprocal-Tail Sequence","stem":"On natural indices starting at zero, u(n)=(5)+(-18)/(n+3), with d≥1. Find the declared reciprocal-tail limit.","answer":{"kind":"short_text","value":"5","explanation":"The reciprocal error tends to zero in this model, leaving L=5, including c=0. This is not an inference about an arbitrary sequence from finitely many terms."},"skills":["Limit of an Explicit Reciprocal-Tail Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0019","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0731206df082a403d346f2ed2871a85a1dc81e4c01c00558ed7e333a38e78c2f","family_id":"qf-sequence-limit-reciprocal-value","seed":202610081019,"semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"6\"],\"operation\":\"reciprocal_sequence_limit\"}"},"fields":{"title":"Limit of an Explicit Reciprocal-Tail Sequence","stem":"On natural indices starting at zero, u(n)=(6)+(19)/(n+4), with d≥1. Find the declared reciprocal-tail limit.","answer":{"kind":"short_text","value":"6","explanation":"The reciprocal error tends to zero in this model, leaving L=6, including c=0. This is not an inference about an arbitrary sequence from finitely many terms."},"skills":["Limit of an Explicit Reciprocal-Tail Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0020","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af100cee1c99cd7048d5f32dc364d2bef06010102ffa77e7666e6b405988d104","family_id":"qf-sequence-limit-reciprocal-value","seed":202610081020,"semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"7\"],\"operation\":\"reciprocal_sequence_limit\"}"},"fields":{"title":"Limit of an Explicit Reciprocal-Tail Sequence","stem":"On natural indices starting at zero, u(n)=(7)+(-20)/(n+5), with d≥1. Find the declared reciprocal-tail limit.","answer":{"kind":"short_text","value":"7","explanation":"The reciprocal error tends to zero in this model, leaving L=7, including c=0. This is not an inference about an arbitrary sequence from finitely many terms."},"skills":["Limit of an Explicit Reciprocal-Tail Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0021","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"907c186455b9b94c04781e4b9bf1c2aa23e74ef047dd0571a141f5695c26df45","family_id":"qf-sequence-limit-reciprocal-value","seed":202610081021,"semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"8\"],\"operation\":\"reciprocal_sequence_limit\"}"},"fields":{"title":"Limit of an Explicit Reciprocal-Tail Sequence","stem":"On natural indices starting at zero, u(n)=(8)+(21)/(n+1), with d≥1. Find the declared reciprocal-tail limit.","answer":{"kind":"short_text","value":"8","explanation":"The reciprocal error tends to zero in this model, leaving L=8, including c=0. This is not an inference about an arbitrary sequence from finitely many terms."},"skills":["Limit of an Explicit Reciprocal-Tail Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0022","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed695f1ec28820cf942bd24d20461b5ef4f45b9284b069b3979ce35e03e04e30","family_id":"qf-sequence-limit-reciprocal-value","seed":202610081022,"semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"9\"],\"operation\":\"reciprocal_sequence_limit\"}"},"fields":{"title":"Limit of an Explicit Reciprocal-Tail Sequence","stem":"On natural indices starting at zero, u(n)=(9)+(-22)/(n+2), with d≥1. Find the declared reciprocal-tail limit.","answer":{"kind":"short_text","value":"9","explanation":"The reciprocal error tends to zero in this model, leaving L=9, including c=0. This is not an inference about an arbitrary sequence from finitely many terms."},"skills":["Limit of an Explicit Reciprocal-Tail Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0023","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19ef6e5c473a6e3d9318608d390a5ade48cee22eb58a78570b97dffb5cd58c3d","family_id":"qf-sequence-limit-reciprocal-value","seed":202610081023,"semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"10\"],\"operation\":\"reciprocal_sequence_limit\"}"},"fields":{"title":"Limit of an Explicit Reciprocal-Tail Sequence","stem":"On natural indices starting at zero, u(n)=(10)+(23)/(n+3), with d≥1. Find the declared reciprocal-tail limit.","answer":{"kind":"short_text","value":"10","explanation":"The reciprocal error tends to zero in this model, leaving L=10, including c=0. This is not an inference about an arbitrary sequence from finitely many terms."},"skills":["Limit of an Explicit Reciprocal-Tail Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0024","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"853258dc63295756db23d18d9f249605f169075a545ed9d9b5c35c36e3cc64f3","family_id":"qf-sequence-limit-reciprocal-value","seed":202610081024,"semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"11\"],\"operation\":\"reciprocal_sequence_limit\"}"},"fields":{"title":"Limit of an Explicit Reciprocal-Tail Sequence","stem":"On natural indices starting at zero, u(n)=(11)+(-24)/(n+4), with d≥1. Find the declared reciprocal-tail limit.","answer":{"kind":"short_text","value":"11","explanation":"The reciprocal error tends to zero in this model, leaving L=11, including c=0. This is not an inference about an arbitrary sequence from finitely many terms."},"skills":["Limit of an Explicit Reciprocal-Tail Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0025","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea8dee71ffe9c18ff3253c81a3ea33099d6cfe75b7f5b3741dd4b639ee689a99","family_id":"qf-sequence-limit-reciprocal-value","seed":202610081025,"semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"12\"],\"operation\":\"reciprocal_sequence_limit\"}"},"fields":{"title":"Limit of an Explicit Reciprocal-Tail Sequence","stem":"On natural indices starting at zero, u(n)=(12)+(25)/(n+5), with d≥1. Find the declared reciprocal-tail limit.","answer":{"kind":"short_text","value":"12","explanation":"The reciprocal error tends to zero in this model, leaving L=12, including c=0. This is not an inference about an arbitrary sequence from finitely many terms."},"skills":["Limit of an Explicit Reciprocal-Tail Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0026","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70095193f621df14214519914b14b90776ab71b89a2b6a3a5a562d3d6777e1ca","family_id":"qf-sequence-limit-least-cutoff","seed":202610081026,"semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"constant_tail\"],\"operation\":\"least_cutoff\"}"},"fields":{"title":"Least Natural Cutoff Including the Cutoff Term","stem":"For an arbitrary fixed real L, u(n)=L+(0)/(n+1) on natural indices starting at zero, with d=1≥1. For ε=1/1>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.","answer":{"kind":"short_text","value":"0","explanation":"The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=0. Equality fails the strict bound; a zero tail permits N=0."},"skills":["Least Natural Cutoff Including the Cutoff Term"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0027","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79491bca067edfbc671c95807779d9a2eac43d78686b175dde1fd111003eb847","family_id":"qf-sequence-limit-least-cutoff","seed":202610081027,"semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"2\",\"2\",\"1/1\"],\"operation\":\"least_cutoff\"}"},"fields":{"title":"Least Natural Cutoff Including the Cutoff Term","stem":"For an arbitrary fixed real L, u(n)=L+(-2)/(n+2) on natural indices starting at zero, with d=2≥1. For ε=2/2>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.","answer":{"kind":"short_text","value":"1","explanation":"The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=1. Equality fails the strict bound; a zero tail permits N=0."},"skills":["Least Natural Cutoff Including the Cutoff Term"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0028","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1dc1606506f9612f3070e8f3ca1682c4e2c283598b428024d05d650a7b3c5904","family_id":"qf-sequence-limit-least-cutoff","seed":202610081028,"semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"3\",\"3\",\"1/1\"],\"operation\":\"least_cutoff\"}"},"fields":{"title":"Least Natural Cutoff Including the Cutoff Term","stem":"For an arbitrary fixed real L, u(n)=L+(3)/(n+3) on natural indices starting at zero, with d=3≥1. For ε=3/3>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.","answer":{"kind":"short_text","value":"1","explanation":"The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=1. Equality fails the strict bound; a zero tail permits N=0."},"skills":["Least Natural Cutoff Including the Cutoff Term"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0029","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4207d1884f1c79dff4a4a8117db4966e1c4cc6db4af8448f3a12c5185a3cb92","family_id":"qf-sequence-limit-least-cutoff","seed":202610081029,"semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"4\",\"4\",\"4/1\"],\"operation\":\"least_cutoff\"}"},"fields":{"title":"Least Natural Cutoff Including the Cutoff Term","stem":"For an arbitrary fixed real L, u(n)=L+(-4)/(n+4) on natural indices starting at zero, with d=4≥1. For ε=4/1>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.","answer":{"kind":"short_text","value":"0","explanation":"The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=0. Equality fails the strict bound; a zero tail permits N=0."},"skills":["Least Natural Cutoff Including the Cutoff Term"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0030","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54b5a04230f462ae9fae9dd1779f93803f5fab9a330a43e2280c08982e16a2b7","family_id":"qf-sequence-limit-least-cutoff","seed":202610081030,"semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"5\",\"1\",\"5/2\"],\"operation\":\"least_cutoff\"}"},"fields":{"title":"Least Natural Cutoff Including the Cutoff Term","stem":"For an arbitrary fixed real L, u(n)=L+(5)/(n+1) on natural indices starting at zero, with d=1≥1. For ε=5/2>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.","answer":{"kind":"short_text","value":"2","explanation":"The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=2. Equality fails the strict bound; a zero tail permits N=0."},"skills":["Least Natural Cutoff Including the Cutoff Term"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0031","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed20de4041923d5398c324f4db6c0aa589939720329bbf8db8d8615c3a8e0b5f","family_id":"qf-sequence-limit-least-cutoff","seed":202610081031,"semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"6\",\"2\",\"1/3\"],\"operation\":\"least_cutoff\"}"},"fields":{"title":"Least Natural Cutoff Including the Cutoff Term","stem":"For an arbitrary fixed real L, u(n)=L+(-6)/(n+2) on natural indices starting at zero, with d=2≥1. For ε=1/3>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.","answer":{"kind":"short_text","value":"17","explanation":"The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=17. Equality fails the strict bound; a zero tail permits N=0."},"skills":["Least Natural Cutoff Including the Cutoff Term"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0032","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2c8eda38d26ac4529a77f0179fe7df472bc8a86dd7eda1c0847ba61d6594db3","family_id":"qf-sequence-limit-least-cutoff","seed":202610081032,"semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"7\",\"3\",\"2/1\"],\"operation\":\"least_cutoff\"}"},"fields":{"title":"Least Natural Cutoff Including the Cutoff Term","stem":"For an arbitrary fixed real L, u(n)=L+(7)/(n+3) on natural indices starting at zero, with d=3≥1. For ε=2/1>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.","answer":{"kind":"short_text","value":"1","explanation":"The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=1. Equality fails the strict bound; a zero tail permits N=0."},"skills":["Least Natural Cutoff Including the Cutoff Term"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0033","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83156d3cd6bb86f3d45e0358a59b3610d9609aa0cf329b23a22b4191f28a37f2","family_id":"qf-sequence-limit-least-cutoff","seed":202610081033,"semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"8\",\"4\",\"3/2\"],\"operation\":\"least_cutoff\"}"},"fields":{"title":"Least Natural Cutoff Including the Cutoff Term","stem":"For an arbitrary fixed real L, u(n)=L+(-8)/(n+4) on natural indices starting at zero, with d=4≥1. For ε=3/2>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.","answer":{"kind":"short_text","value":"2","explanation":"The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=2. Equality fails the strict bound; a zero tail permits N=0."},"skills":["Least Natural Cutoff Including the Cutoff Term"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0034","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5fb2efb9bd6e41fe571fe62cbbe8cd929cc5c13e0eb846e68432b9eeccea4d84","family_id":"qf-sequence-limit-least-cutoff","seed":202610081034,"semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"9\",\"1\",\"4/3\"],\"operation\":\"least_cutoff\"}"},"fields":{"title":"Least Natural Cutoff Including the Cutoff Term","stem":"For an arbitrary fixed real L, u(n)=L+(9)/(n+1) on natural indices starting at zero, with d=1≥1. For ε=4/3>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.","answer":{"kind":"short_text","value":"6","explanation":"The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=6. Equality fails the strict bound; a zero tail permits N=0."},"skills":["Least Natural Cutoff Including the Cutoff Term"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0035","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a16323c9ea6ea6ca80c3d51479cfe1f10ae2fbc827ce979d2ccf71ae4a15aef","family_id":"qf-sequence-limit-least-cutoff","seed":202610081035,"semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"10\",\"2\",\"5/1\"],\"operation\":\"least_cutoff\"}"},"fields":{"title":"Least Natural Cutoff Including the Cutoff Term","stem":"For an arbitrary fixed real L, u(n)=L+(-10)/(n+2) on natural indices starting at zero, with d=2≥1. For ε=5/1>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.","answer":{"kind":"short_text","value":"1","explanation":"The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=1. Equality fails the strict bound; a zero tail permits N=0."},"skills":["Least Natural Cutoff Including the Cutoff Term"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0036","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e257ec3b1c00df8aaff7e414e460626b66f99b92f77b055212eb38712839d750","family_id":"qf-sequence-limit-least-cutoff","seed":202610081036,"semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"11\",\"3\",\"1/2\"],\"operation\":\"least_cutoff\"}"},"fields":{"title":"Least Natural Cutoff Including the Cutoff Term","stem":"For an arbitrary fixed real L, u(n)=L+(11)/(n+3) on natural indices starting at zero, with d=3≥1. For ε=1/2>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.","answer":{"kind":"short_text","value":"20","explanation":"The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=20. Equality fails the strict bound; a zero tail permits N=0."},"skills":["Least Natural Cutoff Including the Cutoff Term"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0037","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d95424664d5f5b7f339588321902498d0c07e39c4331903337ced9dd91acfc7d","family_id":"qf-sequence-limit-least-cutoff","seed":202610081037,"semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"12\",\"4\",\"2/3\"],\"operation\":\"least_cutoff\"}"},"fields":{"title":"Least Natural Cutoff Including the Cutoff Term","stem":"For an arbitrary fixed real L, u(n)=L+(-12)/(n+4) on natural indices starting at zero, with d=4≥1. For ε=2/3>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.","answer":{"kind":"short_text","value":"15","explanation":"The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=15. Equality fails the strict bound; a zero tail permits N=0."},"skills":["Least Natural Cutoff Including the Cutoff Term"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0038","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f496eea520b16569d924b39eda69b9099543430d5d4c10fffeed973bead3ed94","family_id":"qf-sequence-limit-least-cutoff","seed":202610081038,"semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"13\",\"1\",\"3/1\"],\"operation\":\"least_cutoff\"}"},"fields":{"title":"Least Natural Cutoff Including the Cutoff Term","stem":"For an arbitrary fixed real L, u(n)=L+(13)/(n+1) on natural indices starting at zero, with d=1≥1. For ε=3/1>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.","answer":{"kind":"short_text","value":"4","explanation":"The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=4. Equality fails the strict bound; a zero tail permits N=0."},"skills":["Least Natural Cutoff Including the Cutoff Term"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0039","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e02a0e12ba7dd462995c034ce8f17760387e2ae2d7da87e479add0566d3842b","family_id":"qf-sequence-limit-least-cutoff","seed":202610081039,"semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"14\",\"2\",\"2/1\"],\"operation\":\"least_cutoff\"}"},"fields":{"title":"Least Natural Cutoff Including the Cutoff Term","stem":"For an arbitrary fixed real L, u(n)=L+(-14)/(n+2) on natural indices starting at zero, with d=2≥1. For ε=4/2>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.","answer":{"kind":"short_text","value":"6","explanation":"The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=6. Equality fails the strict bound; a zero tail permits N=0."},"skills":["Least Natural Cutoff Including the Cutoff Term"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0040","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5980370ffdb574c4bb8ba644f86219f6aebaf0eb93d43e71ce74ef0ff8ff609e","family_id":"qf-sequence-limit-least-cutoff","seed":202610081040,"semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"15\",\"3\",\"5/3\"],\"operation\":\"least_cutoff\"}"},"fields":{"title":"Least Natural Cutoff Including the Cutoff Term","stem":"For an arbitrary fixed real L, u(n)=L+(15)/(n+3) on natural indices starting at zero, with d=3≥1. For ε=5/3>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.","answer":{"kind":"short_text","value":"7","explanation":"The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=7. Equality fails the strict bound; a zero tail permits N=0."},"skills":["Least Natural Cutoff Including the Cutoff Term"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0041","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f59f554d84f8b3436ae15d9023351c9123055404349486cd79d9008134ed2a91","family_id":"qf-sequence-limit-least-cutoff","seed":202610081041,"semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"16\",\"4\",\"1/1\"],\"operation\":\"least_cutoff\"}"},"fields":{"title":"Least Natural Cutoff Including the Cutoff Term","stem":"For an arbitrary fixed real L, u(n)=L+(-16)/(n+4) on natural indices starting at zero, with d=4≥1. For ε=1/1>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.","answer":{"kind":"short_text","value":"13","explanation":"The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=13. Equality fails the strict bound; a zero tail permits N=0."},"skills":["Least Natural Cutoff Including the Cutoff Term"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0042","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97ec75582eaa0a562fdf72225e21ed7fcd97f49d00a301aefc6011e03c1a8089","family_id":"qf-sequence-limit-least-cutoff","seed":202610081042,"semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"17\",\"1\",\"1/1\"],\"operation\":\"least_cutoff\"}"},"fields":{"title":"Least Natural Cutoff Including the Cutoff Term","stem":"For an arbitrary fixed real L, u(n)=L+(17)/(n+1) on natural indices starting at zero, with d=1≥1. For ε=2/2>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.","answer":{"kind":"short_text","value":"17","explanation":"The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=17. Equality fails the strict bound; a zero tail permits N=0."},"skills":["Least Natural Cutoff Including the Cutoff Term"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0043","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c6d26affd93c79e0c160e9952e12a9a0a095a2e6cfa459f71d97fa705e5e2ce3","family_id":"qf-sequence-limit-least-cutoff","seed":202610081043,"semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"18\",\"2\",\"1/1\"],\"operation\":\"least_cutoff\"}"},"fields":{"title":"Least Natural Cutoff Including the Cutoff Term","stem":"For an arbitrary fixed real L, u(n)=L+(-18)/(n+2) on natural indices starting at zero, with d=2≥1. For ε=3/3>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.","answer":{"kind":"short_text","value":"17","explanation":"The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=17. Equality fails the strict bound; a zero tail permits N=0."},"skills":["Least Natural Cutoff Including the Cutoff Term"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0044","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d1db003f1f5cf2325df893a7e3ce78ff73901a1598f9ec7c0f311b0921600f6","family_id":"qf-sequence-limit-least-cutoff","seed":202610081044,"semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"19\",\"3\",\"4/1\"],\"operation\":\"least_cutoff\"}"},"fields":{"title":"Least Natural Cutoff Including the Cutoff Term","stem":"For an arbitrary fixed real L, u(n)=L+(19)/(n+3) on natural indices starting at zero, with d=3≥1. For ε=4/1>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.","answer":{"kind":"short_text","value":"2","explanation":"The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=2. Equality fails the strict bound; a zero tail permits N=0."},"skills":["Least Natural Cutoff Including the Cutoff Term"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0045","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ba3b5a17dac8637355f668def2768a3541921529d01fbf5826b93e555798497","family_id":"qf-sequence-limit-least-cutoff","seed":202610081045,"semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"20\",\"4\",\"5/2\"],\"operation\":\"least_cutoff\"}"},"fields":{"title":"Least Natural Cutoff Including the Cutoff Term","stem":"For an arbitrary fixed real L, u(n)=L+(-20)/(n+4) on natural indices starting at zero, with d=4≥1. For ε=5/2>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.","answer":{"kind":"short_text","value":"5","explanation":"The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=5. Equality fails the strict bound; a zero tail permits N=0."},"skills":["Least Natural Cutoff Including the Cutoff Term"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0046","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d555957e8d1a565eae575cb79da802a74183a91d1b0f1546b715fad4f9e5b04","family_id":"qf-sequence-limit-least-cutoff","seed":202610081046,"semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"21\",\"1\",\"1/3\"],\"operation\":\"least_cutoff\"}"},"fields":{"title":"Least Natural Cutoff Including the Cutoff Term","stem":"For an arbitrary fixed real L, u(n)=L+(21)/(n+1) on natural indices starting at zero, with d=1≥1. For ε=1/3>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.","answer":{"kind":"short_text","value":"63","explanation":"The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=63. Equality fails the strict bound; a zero tail permits N=0."},"skills":["Least Natural Cutoff Including the Cutoff Term"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0047","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b9c46d25fcbf2f5d4288d0cbb030ef7e59f89e653b07aaacceec8c444135827","family_id":"qf-sequence-limit-least-cutoff","seed":202610081047,"semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"22\",\"2\",\"2/1\"],\"operation\":\"least_cutoff\"}"},"fields":{"title":"Least Natural Cutoff Including the Cutoff Term","stem":"For an arbitrary fixed real L, u(n)=L+(-22)/(n+2) on natural indices starting at zero, with d=2≥1. For ε=2/1>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.","answer":{"kind":"short_text","value":"10","explanation":"The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=10. Equality fails the strict bound; a zero tail permits N=0."},"skills":["Least Natural Cutoff Including the Cutoff Term"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0048","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9394cd95d3d58d496f97d1ee0ece67dee557f1c0be23c7957f89653be3367780","family_id":"qf-sequence-limit-least-cutoff","seed":202610081048,"semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"23\",\"3\",\"3/2\"],\"operation\":\"least_cutoff\"}"},"fields":{"title":"Least Natural Cutoff Including the Cutoff Term","stem":"For an arbitrary fixed real L, u(n)=L+(23)/(n+3) on natural indices starting at zero, with d=3≥1. For ε=3/2>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.","answer":{"kind":"short_text","value":"13","explanation":"The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=13. Equality fails the strict bound; a zero tail permits N=0."},"skills":["Least Natural Cutoff Including the Cutoff Term"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0049","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e9d81dbfaf8f81b5d7bf9aaa61732d7dea73b57ca65d880c8c8b0c964fe1af2","family_id":"qf-sequence-limit-least-cutoff","seed":202610081049,"semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"24\",\"4\",\"4/3\"],\"operation\":\"least_cutoff\"}"},"fields":{"title":"Least Natural Cutoff Including the Cutoff Term","stem":"For an arbitrary fixed real L, u(n)=L+(-24)/(n+4) on natural indices starting at zero, with d=4≥1. For ε=4/3>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.","answer":{"kind":"short_text","value":"15","explanation":"The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=15. Equality fails the strict bound; a zero tail permits N=0."},"skills":["Least Natural Cutoff Including the Cutoff Term"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0050","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2208e2c860057ae3f014d883187c6402c448ec4026f90329e8d6c07c2f9aff7","family_id":"qf-sequence-limit-least-cutoff","seed":202610081050,"semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"25\",\"1\",\"5/1\"],\"operation\":\"least_cutoff\"}"},"fields":{"title":"Least Natural Cutoff Including the Cutoff Term","stem":"For an arbitrary fixed real L, u(n)=L+(25)/(n+1) on natural indices starting at zero, with d=1≥1. For ε=5/1>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.","answer":{"kind":"short_text","value":"5","explanation":"The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=5. Equality fails the strict bound; a zero tail permits N=0."},"skills":["Least Natural Cutoff Including the Cutoff Term"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0051","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6486d3a8338b6edd410aef1e37a411264f7cf9636949c0627265f4960d947ddf","family_id":"qf-sequence-limit-cutoff-validity","seed":202610081051,"semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"constant_tail\"],\"operation\":\"cutoff_validity\"}"},"fields":{"title":"Strict Eligibility of a Candidate Tail Cutoff","stem":"For an arbitrary fixed real L, u(n)=L+(0)/(n+1) on natural indices starting at zero, with d=1≥1. With ε=1/1>0 and candidate N=0, does every n≥N have |u(n)−L|<ε, including n=N?","answer":{"kind":"short_text","value":"Yes","explanation":"Check |c|q<p(N+d) exactly, giving Yes. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence."},"skills":["Strict Eligibility of a Candidate Tail Cutoff"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0052","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"648e28f29ae31931d452c05958ab991a3e8fbddcbf377fa74e698922103f2f1c","family_id":"qf-sequence-limit-cutoff-validity","seed":202610081052,"semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"2\",\"2\",\"1/1\",\"0\"],\"operation\":\"cutoff_validity\"}"},"fields":{"title":"Strict Eligibility of a Candidate Tail Cutoff","stem":"For an arbitrary fixed real L, u(n)=L+(-2)/(n+2) on natural indices starting at zero, with d=2≥1. With ε=2/2>0 and candidate N=0, does every n≥N have |u(n)−L|<ε, including n=N?","answer":{"kind":"short_text","value":"No","explanation":"Check |c|q<p(N+d) exactly, giving No. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence."},"skills":["Strict Eligibility of a Candidate Tail Cutoff"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0053","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fef35467c536bc304e22fb121de5cf330f494ff1599358d1d9ce0a12fc26e4a5","family_id":"qf-sequence-limit-cutoff-validity","seed":202610081053,"semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"3\",\"3\",\"1/1\",\"3\"],\"operation\":\"cutoff_validity\"}"},"fields":{"title":"Strict Eligibility of a Candidate Tail Cutoff","stem":"For an arbitrary fixed real L, u(n)=L+(3)/(n+3) on natural indices starting at zero, with d=3≥1. With ε=3/3>0 and candidate N=3, does every n≥N have |u(n)−L|<ε, including n=N?","answer":{"kind":"short_text","value":"Yes","explanation":"Check |c|q<p(N+d) exactly, giving Yes. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence."},"skills":["Strict Eligibility of a Candidate Tail Cutoff"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0054","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf310970617b2f749bae40c4e0855bb01793ab52ac9513dd2adcef0c4f4a48ad","family_id":"qf-sequence-limit-cutoff-validity","seed":202610081054,"semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"4\",\"4\",\"4/1\",\"0\"],\"operation\":\"cutoff_validity\"}"},"fields":{"title":"Strict Eligibility of a Candidate Tail Cutoff","stem":"For an arbitrary fixed real L, u(n)=L+(-4)/(n+4) on natural indices starting at zero, with d=4≥1. With ε=4/1>0 and candidate N=0, does every n≥N have |u(n)−L|<ε, including n=N?","answer":{"kind":"short_text","value":"Yes","explanation":"Check |c|q<p(N+d) exactly, giving Yes. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence."},"skills":["Strict Eligibility of a Candidate Tail Cutoff"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0055","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7cca54ec4bc8f025d8e89e15dc49e216ad299102689e27c76459e750591fce68","family_id":"qf-sequence-limit-cutoff-validity","seed":202610081055,"semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"5\",\"1\",\"5/2\",\"1\"],\"operation\":\"cutoff_validity\"}"},"fields":{"title":"Strict Eligibility of a Candidate Tail Cutoff","stem":"For an arbitrary fixed real L, u(n)=L+(5)/(n+1) on natural indices starting at zero, with d=1≥1. With ε=5/2>0 and candidate N=1, does every n≥N have |u(n)−L|<ε, including n=N?","answer":{"kind":"short_text","value":"No","explanation":"Check |c|q<p(N+d) exactly, giving No. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence."},"skills":["Strict Eligibility of a Candidate Tail Cutoff"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0056","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9a47f47cb47deddea0b63427030c19916ae530cc0879b4135a1b82dc56c6b8e","family_id":"qf-sequence-limit-cutoff-validity","seed":202610081056,"semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"6\",\"2\",\"1/3\",\"19\"],\"operation\":\"cutoff_validity\"}"},"fields":{"title":"Strict Eligibility of a Candidate Tail Cutoff","stem":"For an arbitrary fixed real L, u(n)=L+(-6)/(n+2) on natural indices starting at zero, with d=2≥1. With ε=1/3>0 and candidate N=19, does every n≥N have |u(n)−L|<ε, including n=N?","answer":{"kind":"short_text","value":"Yes","explanation":"Check |c|q<p(N+d) exactly, giving Yes. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence."},"skills":["Strict Eligibility of a Candidate Tail Cutoff"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0057","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c9422db9fcec9f777daa60d1e11e83aa6ba947869246fbb8e216a889bdfc35f","family_id":"qf-sequence-limit-cutoff-validity","seed":202610081057,"semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"7\",\"3\",\"2/1\",\"1\"],\"operation\":\"cutoff_validity\"}"},"fields":{"title":"Strict Eligibility of a Candidate Tail Cutoff","stem":"For an arbitrary fixed real L, u(n)=L+(7)/(n+3) on natural indices starting at zero, with d=3≥1. With ε=2/1>0 and candidate N=1, does every n≥N have |u(n)−L|<ε, including n=N?","answer":{"kind":"short_text","value":"Yes","explanation":"Check |c|q<p(N+d) exactly, giving Yes. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence."},"skills":["Strict Eligibility of a Candidate Tail Cutoff"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0058","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75488a49e6f085f67f83a2faab9870454973022175b0f74fd23e1b0dfcc3fb43","family_id":"qf-sequence-limit-cutoff-validity","seed":202610081058,"semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"8\",\"4\",\"3/2\",\"1\"],\"operation\":\"cutoff_validity\"}"},"fields":{"title":"Strict Eligibility of a Candidate Tail Cutoff","stem":"For an arbitrary fixed real L, u(n)=L+(-8)/(n+4) on natural indices starting at zero, with d=4≥1. With ε=3/2>0 and candidate N=1, does every n≥N have |u(n)−L|<ε, including n=N?","answer":{"kind":"short_text","value":"No","explanation":"Check |c|q<p(N+d) exactly, giving No. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence."},"skills":["Strict Eligibility of a Candidate Tail Cutoff"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0059","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70d3af0e2343f97d708d8d6231231176ab17f590c929a769530cfe46c84e1bfb","family_id":"qf-sequence-limit-cutoff-validity","seed":202610081059,"semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"9\",\"1\",\"4/3\",\"8\"],\"operation\":\"cutoff_validity\"}"},"fields":{"title":"Strict Eligibility of a Candidate Tail Cutoff","stem":"For an arbitrary fixed real L, u(n)=L+(9)/(n+1) on natural indices starting at zero, with d=1≥1. With ε=4/3>0 and candidate N=8, does every n≥N have |u(n)−L|<ε, including n=N?","answer":{"kind":"short_text","value":"Yes","explanation":"Check |c|q<p(N+d) exactly, giving Yes. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence."},"skills":["Strict Eligibility of a Candidate Tail Cutoff"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0060","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4a96aed8005c8eb9dcfaa2fe6b953af016e91d682da4da821279bfc6b2506e2","family_id":"qf-sequence-limit-cutoff-validity","seed":202610081060,"semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"10\",\"2\",\"5/1\",\"1\"],\"operation\":\"cutoff_validity\"}"},"fields":{"title":"Strict Eligibility of a Candidate Tail Cutoff","stem":"For an arbitrary fixed real L, u(n)=L+(-10)/(n+2) on natural indices starting at zero, with d=2≥1. With ε=5/1>0 and candidate N=1, does every n≥N have |u(n)−L|<ε, including n=N?","answer":{"kind":"short_text","value":"Yes","explanation":"Check |c|q<p(N+d) exactly, giving Yes. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence."},"skills":["Strict Eligibility of a Candidate Tail Cutoff"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0061","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c1503a781e0598780866111aeed0dad02120ab76fe4ab39d5635a2c18b85b56","family_id":"qf-sequence-limit-cutoff-validity","seed":202610081061,"semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"11\",\"3\",\"1/2\",\"19\"],\"operation\":\"cutoff_validity\"}"},"fields":{"title":"Strict Eligibility of a Candidate Tail Cutoff","stem":"For an arbitrary fixed real L, u(n)=L+(11)/(n+3) on natural indices starting at zero, with d=3≥1. With ε=1/2>0 and candidate N=19, does every n≥N have |u(n)−L|<ε, including n=N?","answer":{"kind":"short_text","value":"No","explanation":"Check |c|q<p(N+d) exactly, giving No. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence."},"skills":["Strict Eligibility of a Candidate Tail Cutoff"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0062","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"782f3f397f74842b2084a966f8e79485f0c03c969ac0e8e1946c358d335b6b24","family_id":"qf-sequence-limit-cutoff-validity","seed":202610081062,"semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"12\",\"4\",\"2/3\",\"17\"],\"operation\":\"cutoff_validity\"}"},"fields":{"title":"Strict Eligibility of a Candidate Tail Cutoff","stem":"For an arbitrary fixed real L, u(n)=L+(-12)/(n+4) on natural indices starting at zero, with d=4≥1. With ε=2/3>0 and candidate N=17, does every n≥N have |u(n)−L|<ε, including n=N?","answer":{"kind":"short_text","value":"Yes","explanation":"Check |c|q<p(N+d) exactly, giving Yes. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence."},"skills":["Strict Eligibility of a Candidate Tail Cutoff"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0063","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4b2dbb32b404f01a9576cc675b56f31268d8617a7f85b164e1a30456ca4b2b8","family_id":"qf-sequence-limit-cutoff-validity","seed":202610081063,"semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"13\",\"1\",\"3/1\",\"4\"],\"operation\":\"cutoff_validity\"}"},"fields":{"title":"Strict Eligibility of a Candidate Tail Cutoff","stem":"For an arbitrary fixed real L, u(n)=L+(13)/(n+1) on natural indices starting at zero, with d=1≥1. With ε=3/1>0 and candidate N=4, does every n≥N have |u(n)−L|<ε, including n=N?","answer":{"kind":"short_text","value":"Yes","explanation":"Check |c|q<p(N+d) exactly, giving Yes. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence."},"skills":["Strict Eligibility of a Candidate Tail Cutoff"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0064","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b461557e95879d83c06b8cbb14044888e81c1127d7766841f3ce2bc5f49f6b8a","family_id":"qf-sequence-limit-cutoff-validity","seed":202610081064,"semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"14\",\"2\",\"2/1\",\"5\"],\"operation\":\"cutoff_validity\"}"},"fields":{"title":"Strict Eligibility of a Candidate Tail Cutoff","stem":"For an arbitrary fixed real L, u(n)=L+(-14)/(n+2) on natural indices starting at zero, with d=2≥1. With ε=4/2>0 and candidate N=5, does every n≥N have |u(n)−L|<ε, including n=N?","answer":{"kind":"short_text","value":"No","explanation":"Check |c|q<p(N+d) exactly, giving No. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence."},"skills":["Strict Eligibility of a Candidate Tail Cutoff"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0065","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0865602b54679d52212e014de4dda07580092d15539ed840eaacc8a1892695eb","family_id":"qf-sequence-limit-cutoff-validity","seed":202610081065,"semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"15\",\"3\",\"5/3\",\"9\"],\"operation\":\"cutoff_validity\"}"},"fields":{"title":"Strict Eligibility of a Candidate Tail Cutoff","stem":"For an arbitrary fixed real L, u(n)=L+(15)/(n+3) on natural indices starting at zero, with d=3≥1. With ε=5/3>0 and candidate N=9, does every n≥N have |u(n)−L|<ε, including n=N?","answer":{"kind":"short_text","value":"Yes","explanation":"Check |c|q<p(N+d) exactly, giving Yes. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence."},"skills":["Strict Eligibility of a Candidate Tail Cutoff"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0066","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d588507685415b3652f79d0db557f14fb1594ce983a9983986cb1715ec1e7f2e","family_id":"qf-sequence-limit-cutoff-validity","seed":202610081066,"semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"16\",\"4\",\"1/1\",\"13\"],\"operation\":\"cutoff_validity\"}"},"fields":{"title":"Strict Eligibility of a Candidate Tail Cutoff","stem":"For an arbitrary fixed real L, u(n)=L+(-16)/(n+4) on natural indices starting at zero, with d=4≥1. With ε=1/1>0 and candidate N=13, does every n≥N have |u(n)−L|<ε, including n=N?","answer":{"kind":"short_text","value":"Yes","explanation":"Check |c|q<p(N+d) exactly, giving Yes. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence."},"skills":["Strict Eligibility of a Candidate Tail Cutoff"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0067","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e966e9cf44d0e78eeb51e4d2692a732f04588b537cd8738aec28bb3a6efaeace","family_id":"qf-sequence-limit-cutoff-validity","seed":202610081067,"semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"17\",\"1\",\"1/1\",\"16\"],\"operation\":\"cutoff_validity\"}"},"fields":{"title":"Strict Eligibility of a Candidate Tail Cutoff","stem":"For an arbitrary fixed real L, u(n)=L+(17)/(n+1) on natural indices starting at zero, with d=1≥1. With ε=2/2>0 and candidate N=16, does every n≥N have |u(n)−L|<ε, including n=N?","answer":{"kind":"short_text","value":"No","explanation":"Check |c|q<p(N+d) exactly, giving No. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence."},"skills":["Strict Eligibility of a Candidate Tail Cutoff"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0068","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53bd17a30277feae0ad413e75ae39d95b1b56caa1d66108efdbeb77982aa4f09","family_id":"qf-sequence-limit-cutoff-validity","seed":202610081068,"semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"18\",\"2\",\"1/1\",\"19\"],\"operation\":\"cutoff_validity\"}"},"fields":{"title":"Strict Eligibility of a Candidate Tail Cutoff","stem":"For an arbitrary fixed real L, u(n)=L+(-18)/(n+2) on natural indices starting at zero, with d=2≥1. With ε=3/3>0 and candidate N=19, does every n≥N have |u(n)−L|<ε, including n=N?","answer":{"kind":"short_text","value":"Yes","explanation":"Check |c|q<p(N+d) exactly, giving Yes. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence."},"skills":["Strict Eligibility of a Candidate Tail Cutoff"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0069","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4cc3104724ac62f19263a2a319d0af647bb4bb8c76cf11500856bdcf8ffc2f84","family_id":"qf-sequence-limit-cutoff-validity","seed":202610081069,"semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"19\",\"3\",\"4/1\",\"2\"],\"operation\":\"cutoff_validity\"}"},"fields":{"title":"Strict Eligibility of a Candidate Tail Cutoff","stem":"For an arbitrary fixed real L, u(n)=L+(19)/(n+3) on natural indices starting at zero, with d=3≥1. With ε=4/1>0 and candidate N=2, does every n≥N have |u(n)−L|<ε, including n=N?","answer":{"kind":"short_text","value":"Yes","explanation":"Check |c|q<p(N+d) exactly, giving Yes. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence."},"skills":["Strict Eligibility of a Candidate Tail Cutoff"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0070","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8db2113d061f5082124a56a1b4d333944d23429894bd2379d70f98b01cdcbf3e","family_id":"qf-sequence-limit-cutoff-validity","seed":202610081070,"semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"20\",\"4\",\"5/2\",\"4\"],\"operation\":\"cutoff_validity\"}"},"fields":{"title":"Strict Eligibility of a Candidate Tail Cutoff","stem":"For an arbitrary fixed real L, u(n)=L+(-20)/(n+4) on natural indices starting at zero, with d=4≥1. With ε=5/2>0 and candidate N=4, does every n≥N have |u(n)−L|<ε, including n=N?","answer":{"kind":"short_text","value":"No","explanation":"Check |c|q<p(N+d) exactly, giving No. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence."},"skills":["Strict Eligibility of a Candidate Tail Cutoff"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0071","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e6dbacdd8fe4011344534067074f3c0072f7e7b9515dda074709fdd46eae00b","family_id":"qf-sequence-limit-cutoff-validity","seed":202610081071,"semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"21\",\"1\",\"1/3\",\"65\"],\"operation\":\"cutoff_validity\"}"},"fields":{"title":"Strict Eligibility of a Candidate Tail Cutoff","stem":"For an arbitrary fixed real L, u(n)=L+(21)/(n+1) on natural indices starting at zero, with d=1≥1. With ε=1/3>0 and candidate N=65, does every n≥N have |u(n)−L|<ε, including n=N?","answer":{"kind":"short_text","value":"Yes","explanation":"Check |c|q<p(N+d) exactly, giving Yes. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence."},"skills":["Strict Eligibility of a Candidate Tail Cutoff"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0072","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fbc8823ad97b453b9872f412565575204a3d51f1af616384ca02334c38cf147e","family_id":"qf-sequence-limit-cutoff-validity","seed":202610081072,"semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"22\",\"2\",\"2/1\",\"10\"],\"operation\":\"cutoff_validity\"}"},"fields":{"title":"Strict Eligibility of a Candidate Tail Cutoff","stem":"For an arbitrary fixed real L, u(n)=L+(-22)/(n+2) on natural indices starting at zero, with d=2≥1. With ε=2/1>0 and candidate N=10, does every n≥N have |u(n)−L|<ε, including n=N?","answer":{"kind":"short_text","value":"Yes","explanation":"Check |c|q<p(N+d) exactly, giving Yes. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence."},"skills":["Strict Eligibility of a Candidate Tail Cutoff"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0073","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6214653b856d07bc65f7a0d7b8a54585154bbf824079a1c3f97832c012040955","family_id":"qf-sequence-limit-cutoff-validity","seed":202610081073,"semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"23\",\"3\",\"3/2\",\"12\"],\"operation\":\"cutoff_validity\"}"},"fields":{"title":"Strict Eligibility of a Candidate Tail Cutoff","stem":"For an arbitrary fixed real L, u(n)=L+(23)/(n+3) on natural indices starting at zero, with d=3≥1. With ε=3/2>0 and candidate N=12, does every n≥N have |u(n)−L|<ε, including n=N?","answer":{"kind":"short_text","value":"No","explanation":"Check |c|q<p(N+d) exactly, giving No. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence."},"skills":["Strict Eligibility of a Candidate Tail Cutoff"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0074","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17f556ce575098718127beb17788c380af5fb981056048a3f19050ea3dad6a85","family_id":"qf-sequence-limit-cutoff-validity","seed":202610081074,"semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"24\",\"4\",\"4/3\",\"17\"],\"operation\":\"cutoff_validity\"}"},"fields":{"title":"Strict Eligibility of a Candidate Tail Cutoff","stem":"For an arbitrary fixed real L, u(n)=L+(-24)/(n+4) on natural indices starting at zero, with d=4≥1. With ε=4/3>0 and candidate N=17, does every n≥N have |u(n)−L|<ε, including n=N?","answer":{"kind":"short_text","value":"Yes","explanation":"Check |c|q<p(N+d) exactly, giving Yes. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence."},"skills":["Strict Eligibility of a Candidate Tail Cutoff"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0075","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c66a5d1d229c3f55249fc09fcf8721791639530208b7e2f91bfb65edf30599a3","family_id":"qf-sequence-limit-cutoff-validity","seed":202610081075,"semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"25\",\"1\",\"5/1\",\"5\"],\"operation\":\"cutoff_validity\"}"},"fields":{"title":"Strict Eligibility of a Candidate Tail Cutoff","stem":"For an arbitrary fixed real L, u(n)=L+(25)/(n+1) on natural indices starting at zero, with d=1≥1. With ε=5/1>0 and candidate N=5, does every n≥N have |u(n)−L|<ε, including n=N?","answer":{"kind":"short_text","value":"Yes","explanation":"Check |c|q<p(N+d) exactly, giving Yes. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence."},"skills":["Strict Eligibility of a Candidate Tail Cutoff"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0076","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab81a181d5779a1dec8b947a8934f460843dda708cf2e08a609e448c6b177859","family_id":"qf-sequence-limit-term-error","seed":202610081076,"semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"0\",\"1\"],\"operation\":\"sequence_term_error\"}"},"fields":{"title":"Exact Error at a Specified Sequence Index","stem":"For an arbitrary fixed real L, u(n)=L+(0)/(n+1) on natural indices starting at zero, with d=1≥1. Find the exact absolute error |u(0)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"0/1","explanation":"Use |c|/(n+d) to obtain 0/1. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error."},"skills":["Exact Error at a Specified Sequence Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0077","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a60a1095caa364fc18e33066c58b9ed60fd8a301755195ff83d999a7099ce12a","family_id":"qf-sequence-limit-term-error","seed":202610081077,"semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"2\",\"3\"],\"operation\":\"sequence_term_error\"}"},"fields":{"title":"Exact Error at a Specified Sequence Index","stem":"For an arbitrary fixed real L, u(n)=L+(-2)/(n+2) on natural indices starting at zero, with d=2≥1. Find the exact absolute error |u(1)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"2/3","explanation":"Use |c|/(n+d) to obtain 2/3. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error."},"skills":["Exact Error at a Specified Sequence Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0078","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3075a92d362d3608e439abeea7b71e2a29aa83edbcd42d1a77c806c677d659ea","family_id":"qf-sequence-limit-term-error","seed":202610081078,"semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"3\",\"5\"],\"operation\":\"sequence_term_error\"}"},"fields":{"title":"Exact Error at a Specified Sequence Index","stem":"For an arbitrary fixed real L, u(n)=L+(3)/(n+3) on natural indices starting at zero, with d=3≥1. Find the exact absolute error |u(2)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"3/5","explanation":"Use |c|/(n+d) to obtain 3/5. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error."},"skills":["Exact Error at a Specified Sequence Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0079","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0704c86916a5f4f0e5f8a5cad1b44be44dfc729ef3d5a2ec331f87ed2262c65b","family_id":"qf-sequence-limit-term-error","seed":202610081079,"semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"4\",\"7\"],\"operation\":\"sequence_term_error\"}"},"fields":{"title":"Exact Error at a Specified Sequence Index","stem":"For an arbitrary fixed real L, u(n)=L+(-4)/(n+4) on natural indices starting at zero, with d=4≥1. Find the exact absolute error |u(3)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"4/7","explanation":"Use |c|/(n+d) to obtain 4/7. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error."},"skills":["Exact Error at a Specified Sequence Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0080","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d4e8b3c1241f44e78559d70b7ff07490c56e1064b3cf5b07238a1b66da47db8","family_id":"qf-sequence-limit-term-error","seed":202610081080,"semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"5\",\"5\"],\"operation\":\"sequence_term_error\"}"},"fields":{"title":"Exact Error at a Specified Sequence Index","stem":"For an arbitrary fixed real L, u(n)=L+(5)/(n+1) on natural indices starting at zero, with d=1≥1. Find the exact absolute error |u(4)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"1/1","explanation":"Use |c|/(n+d) to obtain 1/1. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error."},"skills":["Exact Error at a Specified Sequence Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0081","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ab999c28b552e1267e56828479efefd831bfef8ccd1553bf430951ce8e5ee6b","family_id":"qf-sequence-limit-term-error","seed":202610081081,"semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"6\",\"7\"],\"operation\":\"sequence_term_error\"}"},"fields":{"title":"Exact Error at a Specified Sequence Index","stem":"For an arbitrary fixed real L, u(n)=L+(-6)/(n+2) on natural indices starting at zero, with d=2≥1. Find the exact absolute error |u(5)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"6/7","explanation":"Use |c|/(n+d) to obtain 6/7. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error."},"skills":["Exact Error at a Specified Sequence Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0082","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6be605b959ba7fe9f6bc9e16acf443e338705230e27a284a953ae6f6b420e93","family_id":"qf-sequence-limit-term-error","seed":202610081082,"semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"7\",\"9\"],\"operation\":\"sequence_term_error\"}"},"fields":{"title":"Exact Error at a Specified Sequence Index","stem":"For an arbitrary fixed real L, u(n)=L+(7)/(n+3) on natural indices starting at zero, with d=3≥1. Find the exact absolute error |u(6)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"7/9","explanation":"Use |c|/(n+d) to obtain 7/9. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error."},"skills":["Exact Error at a Specified Sequence Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0083","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9629430b4080e7280759020823b4c04d140c97ede2635bb78b3341b0561616b5","family_id":"qf-sequence-limit-term-error","seed":202610081083,"semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"8\",\"4\"],\"operation\":\"sequence_term_error\"}"},"fields":{"title":"Exact Error at a Specified Sequence Index","stem":"For an arbitrary fixed real L, u(n)=L+(-8)/(n+4) on natural indices starting at zero, with d=4≥1. Find the exact absolute error |u(0)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"2/1","explanation":"Use |c|/(n+d) to obtain 2/1. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error."},"skills":["Exact Error at a Specified Sequence Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0084","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"56a7c09465b3053f94b9760087ae20620e7e48d6c429ae35d2c80375d98fdb7d","family_id":"qf-sequence-limit-term-error","seed":202610081084,"semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"9\",\"2\"],\"operation\":\"sequence_term_error\"}"},"fields":{"title":"Exact Error at a Specified Sequence Index","stem":"For an arbitrary fixed real L, u(n)=L+(9)/(n+1) on natural indices starting at zero, with d=1≥1. Find the exact absolute error |u(1)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"9/2","explanation":"Use |c|/(n+d) to obtain 9/2. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error."},"skills":["Exact Error at a Specified Sequence Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0085","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"faea314ab7c5d1beb05f1fb499672d9bea42eb22416c5aa61bc8a6d262a03250","family_id":"qf-sequence-limit-term-error","seed":202610081085,"semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"10\",\"4\"],\"operation\":\"sequence_term_error\"}"},"fields":{"title":"Exact Error at a Specified Sequence Index","stem":"For an arbitrary fixed real L, u(n)=L+(-10)/(n+2) on natural indices starting at zero, with d=2≥1. Find the exact absolute error |u(2)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"5/2","explanation":"Use |c|/(n+d) to obtain 5/2. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error."},"skills":["Exact Error at a Specified Sequence Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0086","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1d30d4b9592beb33e7d584e0c80ef8d8971ce09ce1a56fa97e1a2d964b8c133","family_id":"qf-sequence-limit-term-error","seed":202610081086,"semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"11\",\"6\"],\"operation\":\"sequence_term_error\"}"},"fields":{"title":"Exact Error at a Specified Sequence Index","stem":"For an arbitrary fixed real L, u(n)=L+(11)/(n+3) on natural indices starting at zero, with d=3≥1. Find the exact absolute error |u(3)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"11/6","explanation":"Use |c|/(n+d) to obtain 11/6. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error."},"skills":["Exact Error at a Specified Sequence Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0087","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb6e226f84f20aeebaee9a8ad30845f7eae28d64700543dcd10bbfcce89eafdd","family_id":"qf-sequence-limit-term-error","seed":202610081087,"semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"12\",\"8\"],\"operation\":\"sequence_term_error\"}"},"fields":{"title":"Exact Error at a Specified Sequence Index","stem":"For an arbitrary fixed real L, u(n)=L+(-12)/(n+4) on natural indices starting at zero, with d=4≥1. Find the exact absolute error |u(4)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"3/2","explanation":"Use |c|/(n+d) to obtain 3/2. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error."},"skills":["Exact Error at a Specified Sequence Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0088","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a4127c99e20f00fd799079ce6e083c388c81703b283048accfe51ab2a69cd61","family_id":"qf-sequence-limit-term-error","seed":202610081088,"semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"13\",\"6\"],\"operation\":\"sequence_term_error\"}"},"fields":{"title":"Exact Error at a Specified Sequence Index","stem":"For an arbitrary fixed real L, u(n)=L+(13)/(n+1) on natural indices starting at zero, with d=1≥1. Find the exact absolute error |u(5)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"13/6","explanation":"Use |c|/(n+d) to obtain 13/6. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error."},"skills":["Exact Error at a Specified Sequence Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0089","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1662ed5c5d72628466a474cc751b5bb950ed4486c6f0147fffd1d4a614184d6b","family_id":"qf-sequence-limit-term-error","seed":202610081089,"semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"14\",\"8\"],\"operation\":\"sequence_term_error\"}"},"fields":{"title":"Exact Error at a Specified Sequence Index","stem":"For an arbitrary fixed real L, u(n)=L+(-14)/(n+2) on natural indices starting at zero, with d=2≥1. Find the exact absolute error |u(6)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"7/4","explanation":"Use |c|/(n+d) to obtain 7/4. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error."},"skills":["Exact Error at a Specified Sequence Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0090","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cfd6cfffd9b508a2c2ac932d9944e6719c688998844bf9924e4b18a4311f1ea6","family_id":"qf-sequence-limit-term-error","seed":202610081090,"semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"15\",\"3\"],\"operation\":\"sequence_term_error\"}"},"fields":{"title":"Exact Error at a Specified Sequence Index","stem":"For an arbitrary fixed real L, u(n)=L+(15)/(n+3) on natural indices starting at zero, with d=3≥1. Find the exact absolute error |u(0)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"5/1","explanation":"Use |c|/(n+d) to obtain 5/1. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error."},"skills":["Exact Error at a Specified Sequence Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0091","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43da441b709e1eabb59d23b1fffcc21fba037bf5673b4ee1d272452cce3412aa","family_id":"qf-sequence-limit-term-error","seed":202610081091,"semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"16\",\"5\"],\"operation\":\"sequence_term_error\"}"},"fields":{"title":"Exact Error at a Specified Sequence Index","stem":"For an arbitrary fixed real L, u(n)=L+(-16)/(n+4) on natural indices starting at zero, with d=4≥1. Find the exact absolute error |u(1)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"16/5","explanation":"Use |c|/(n+d) to obtain 16/5. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error."},"skills":["Exact Error at a Specified Sequence Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0092","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62be25754d4711bf01b5e738decf42b3acb3d479cb597537e709761ed72f3ef6","family_id":"qf-sequence-limit-term-error","seed":202610081092,"semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"17\",\"3\"],\"operation\":\"sequence_term_error\"}"},"fields":{"title":"Exact Error at a Specified Sequence Index","stem":"For an arbitrary fixed real L, u(n)=L+(17)/(n+1) on natural indices starting at zero, with d=1≥1. Find the exact absolute error |u(2)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"17/3","explanation":"Use |c|/(n+d) to obtain 17/3. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error."},"skills":["Exact Error at a Specified Sequence Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0093","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65a6ab52b3f60a333a8a873cb2adcda13ba4d3df6976e5b43b9acf173253c26e","family_id":"qf-sequence-limit-term-error","seed":202610081093,"semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"18\",\"5\"],\"operation\":\"sequence_term_error\"}"},"fields":{"title":"Exact Error at a Specified Sequence Index","stem":"For an arbitrary fixed real L, u(n)=L+(-18)/(n+2) on natural indices starting at zero, with d=2≥1. Find the exact absolute error |u(3)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"18/5","explanation":"Use |c|/(n+d) to obtain 18/5. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error."},"skills":["Exact Error at a Specified Sequence Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0094","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"020e2ee55d5520e3d82ab1b23d1c246df973b05f1da7376de3e2a3911786fe13","family_id":"qf-sequence-limit-term-error","seed":202610081094,"semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"19\",\"7\"],\"operation\":\"sequence_term_error\"}"},"fields":{"title":"Exact Error at a Specified Sequence Index","stem":"For an arbitrary fixed real L, u(n)=L+(19)/(n+3) on natural indices starting at zero, with d=3≥1. Find the exact absolute error |u(4)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"19/7","explanation":"Use |c|/(n+d) to obtain 19/7. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error."},"skills":["Exact Error at a Specified Sequence Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0095","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8f75f6469515c342aacae5990df379b1c2fb4a428b3851d367397058921341c","family_id":"qf-sequence-limit-term-error","seed":202610081095,"semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"20\",\"9\"],\"operation\":\"sequence_term_error\"}"},"fields":{"title":"Exact Error at a Specified Sequence Index","stem":"For an arbitrary fixed real L, u(n)=L+(-20)/(n+4) on natural indices starting at zero, with d=4≥1. Find the exact absolute error |u(5)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"20/9","explanation":"Use |c|/(n+d) to obtain 20/9. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error."},"skills":["Exact Error at a Specified Sequence Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0096","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a15cc3548fb32e80bb1d8b8a050c35384e7aa4e540c3e5b479d31a102aec02d7","family_id":"qf-sequence-limit-term-error","seed":202610081096,"semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"21\",\"7\"],\"operation\":\"sequence_term_error\"}"},"fields":{"title":"Exact Error at a Specified Sequence Index","stem":"For an arbitrary fixed real L, u(n)=L+(21)/(n+1) on natural indices starting at zero, with d=1≥1. Find the exact absolute error |u(6)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"3/1","explanation":"Use |c|/(n+d) to obtain 3/1. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error."},"skills":["Exact Error at a Specified Sequence Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0097","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"578efab7474dbb28e648ba51e0476ba7eb90a74fcf12092a523b0b25f3ee956e","family_id":"qf-sequence-limit-term-error","seed":202610081097,"semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"22\",\"2\"],\"operation\":\"sequence_term_error\"}"},"fields":{"title":"Exact Error at a Specified Sequence Index","stem":"For an arbitrary fixed real L, u(n)=L+(-22)/(n+2) on natural indices starting at zero, with d=2≥1. Find the exact absolute error |u(0)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"11/1","explanation":"Use |c|/(n+d) to obtain 11/1. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error."},"skills":["Exact Error at a Specified Sequence Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0098","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83761ce5ca1103d852ac65aaef35516c176fc720fd6e250e35ab309e084a9f61","family_id":"qf-sequence-limit-term-error","seed":202610081098,"semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"23\",\"4\"],\"operation\":\"sequence_term_error\"}"},"fields":{"title":"Exact Error at a Specified Sequence Index","stem":"For an arbitrary fixed real L, u(n)=L+(23)/(n+3) on natural indices starting at zero, with d=3≥1. Find the exact absolute error |u(1)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"23/4","explanation":"Use |c|/(n+d) to obtain 23/4. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error."},"skills":["Exact Error at a Specified Sequence Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0099","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be442391bd2a58c9985c3d92cf8c1afd0918486dd0c3e5326b5fe1ad086b1d45","family_id":"qf-sequence-limit-term-error","seed":202610081099,"semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"24\",\"6\"],\"operation\":\"sequence_term_error\"}"},"fields":{"title":"Exact Error at a Specified Sequence Index","stem":"For an arbitrary fixed real L, u(n)=L+(-24)/(n+4) on natural indices starting at zero, with d=4≥1. Find the exact absolute error |u(2)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"4/1","explanation":"Use |c|/(n+d) to obtain 4/1. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error."},"skills":["Exact Error at a Specified Sequence Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r01-sequence-limit-0100","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fcb2dc3d3917499642637d7c549df6d6e9164b243778f3ba2ab9045d58d905c6","family_id":"qf-sequence-limit-term-error","seed":202610081100,"semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"25\",\"4\"],\"operation\":\"sequence_term_error\"}"},"fields":{"title":"Exact Error at a Specified Sequence Index","stem":"For an arbitrary fixed real L, u(n)=L+(25)/(n+1) on natural indices starting at zero, with d=1≥1. Find the exact absolute error |u(3)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"25/4","explanation":"Use |c|/(n+d) to obtain 25/4. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error."},"skills":["Exact Error at a Specified Sequence Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}