R0034-17955 · q-20261014-advance-20261104-r03-absolute-convergence-of-series-0005
partial sums與absolute-series exact convergence certificate。操作=absseries-geometric；資料={"a":[5,6],"r":[-1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。
G001 — partial sums與absolute-series exact convergence certificate。操作=absseries-geometric；資料={"a":[5,6],"r":[-1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。
操作前
source operation=absseries-geometric;original question snapshot retained;source presentation interface resolved in this page
操作後
EXPLICIT ordered output fields=[a, r, absolute_series_sum, absolutely_convergent, certificate]
條件：無
原題快照保留歷史介面標記，這裡直接列完整欄位，不需瀏覽器JavaScript或外部查表。欄位名只是presentation metadata，不是數學運算；term-zero的第四個否是『條件是否充分』，不是誤標成另一個converges旗標。
規則：absseries_explicit_source_interface_fields_without_javascript · 依賴：G001
操作前
absolute geometric series one minus effective ratio := sub(1, 1/2)
操作後
absolute geometric series one minus effective ratio = 1/2
條件：無
依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。
規則：exact_rational_arithmetic · 依賴：G001, S001
操作前
absolute geometric series exact infinite candidate sum := div(5/6, 1/2)
操作後
absolute geometric series exact infinite candidate sum = 5/3
條件：無
依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。
規則：exact_rational_arithmetic · 依賴：G001, S002
操作前
absolute geometric series:original SIGNED a=5/6;SIGNED r=-1/2;absolute ratio=1/2<1;effective terms=(5/6)*(1/2)^k,k=0,1,...
操作後
candidate sum=5/3;effective denominator1-ratio=1/2 is nonzero
條件：無
先保留原a、r的符號，依本題是否求絕對值級數才換有效係數及ratio；|r|<1是這份證書域，不用有限樣本或另種零係數例外改寫題目。
規則：series_geometric_exact_input_roles_and_absolute_ratio_domain · 依賴：G001, S003
操作前
absolute geometric series:for ANY N>=1,the first N terms start at k0 with leading=5/6,ratio=1/2
操作後
(1-ratio)*S_N=(5/6)*(1-(1/2)^N);S_N=(5/6)*(1-(1/2)^N)/(1/2);candidate minus S_N=(5/6)*(1/2)^N/(1/2)
條件：無
把有限和乘(1−ratio)，相鄰冪次抵消後只留首尾；這個恆等式對任意N≥1，並不是看幾個partial sums猜公式。
規則：series_geometric_all_N_finite_cancellation_identity · 依賴：G001, S004
操作前
absolute geometric series:|effective ratio|=1/2<1;remainder magnitude=abs(5/6)*abs(1/2)^N/abs(1/2)
操作後
geometric power tends to0;remainder tends to0;partial sums converge to the exact candidate
條件：無
嚴格小於1的幾何冪趨零，固定有限係數與非零分母保餘項趨零，因此partial-sum sequence才有此極限。
規則：series_geometric_remainder_limit_not_finite_prefix_inference · 依賴：G001, S005
操作前
original signed a=5/6;r=-1/2;for EVERY k>=0,abs(a*r^k)=abs(a)*abs(r)^k;absolute-series sum=5/3
操作後
for every m>n>=1,absolute signed tail<=absolute-series tail=abs(a)*abs(r)^n/(1-abs(r)) tending to0;Cauchy criterion and real completeness give convergence of original series
條件：無
絕對值作用在每一項，得到|a|/(1−|r|)，不是|a/(1−r)|。任意有號尾和由絕對尾和控制至0，Cauchy與實數完備性才給原級數收斂，並非有限prefix外推。
規則：absseries_absolute_convergence_implies_original_via_cauchy_tail · 依賴：G001, S006
S007 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S006 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S005 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S004 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S003 · premise_and_result_audit — 以精確有理數重算 absolute geometric series exact infinite candidate sum，不以小數近似替代。S002 · premise_and_result_audit — 以精確有理數重算 absolute geometric series one minus effective ratio，不以小數近似替代。S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。(5/6, -1/2, 5/3, 是, absolute_geometric_certificate)
依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。
覆核日期：2026-10-26
原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 7af45d0c5fc85aa6b482d3833bcb2dd98c74a72284614aa29e0e7adcdc922328candidate record SHA-256: 2e238fcd8f1145b55648312ad8e2fcdce8cae867e0c75a7ade370b460340fe95semantic review receipt SHA-256: f73d099ebe3ff1c69a5d96f08c75b11b9e5b05120af40cf3ef388fce3b5c5336question-first witness SHA-256: 22675ffb6209b4fea076032b29b96e3dc19aad4244c54493f5f65f5e5d875699Lean: NOT_RUN · human review: NOT_PERFORMED