The direction of eventual comparison and finite prefixes

Goal and scope

Prove convergence using geometric upper comparison despite a finite prefix, and show divergent lower comparison.

Every nonnegative tail term after cutoff N=2 is explicit; finite-prefix signs are unrestricted.

Givens and symbols

• algebra
• epsilon
• nonnegative_series

• G1 — a_0=5,a_1=-1; a_n=(3/4)*2^{-n} for n≥2; b_n=2^{-n} for n≥2.

• N=2 — whole-tail cutoff
• A_M — nonnegative-tail partial sum

Rules and sources

• mko_definition · cited — For every sufficiently large n require0≤a_n≤b_n. Convergence of the b series implies convergence of the a series; divergence of the a series implies divergence of the b series. Exact families use geometric tails, arbitrary finite prefixes and positive constant lower bounds. Finitely many terms do not change convergence, but every tail comparison is a universal obligation. Negative terms, a reversed inequality, finitely sampled comparisons or a divergent upper bound do not supply the required test. · mko-series-comparison-test
• algebra · prerequisite — Use real-field identities; division requires a nonzero denominator and multiplication by a positive number preserves inequalities.
• epsilon · prerequisite — Convergence requires for every positive epsilon an integer N controlling every n≥N by a strict error bound; use the Archimedean property of the reals.
• nonnegative_series · prerequisite — Nonnegative partial sums increase; if bounded they converge by real completeness, and if unbounded they diverge; a finite prefix only adds a fixed constant.

Main derivation

n\ge2

0\le a_n=(3/4)2^{-n}\le2^{-n}=b_n

\sum_{n=2}^{M}b_n

\sum_{n=2}^{M}2^{-n}=1/2-2^{-M}\to1/2

A_M=\sum_{n=2}^{M}a_n

0\le A_M\le\sum_{n=2}^{M}b_n\le1/2

A_M\text{ nondecreasing and bounded above}

\sum_{n=2}^{\infty}a_n\text{ converges}

\sum_{n=0}^{M}a_n=5-1+(3/4)\sum_{n=2}^{M}2^{-n}

\sum_{n=0}^{\infty}a_n=4+3/8=35/8

c_n=1,\quad d_n=1+2^{-n}\quad(n\ge2)

0\le c_n\le d_n,\quad\sum_{n=2}^{M}c_n=M-1

\sum_{n=2}^{M}d_n\ge M-1\to\infty

\sum d_n\text{ diverges}

0\le u_n\le v_n\text{ eventually}

\sum v_n\text{ convergent}\Rightarrow\sum u_n\text{ convergent};\quad\sum u_n\text{ divergent}\Rightarrow\sum v_n\text{ divergent}

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · S01, S04, S06, S07 — This is a conditional directional argument, not an extrapolation to unverified comparisons.
• S07 · premise_audit · S06 — A divergent upper bound would give no such conclusion.
• S06 · premise_audit · G1 — The new c,d laws are distinct from the main a,b laws.
• S05 · premise_audit · G1, S02, S04 — Prefix signs are not treated as a tail convergence certificate.
• S04 · premise_audit · S01, S03 — Arbitrary negative tail terms would invalidate this monotonicity proof.
• S03 · premise_audit · S01, S02 — The inequality covers every M≥2, not a finite grid.
• S02 · premise_audit · G1 — Reference convergence is not guessed from the compared series.
• S01 · premise_audit · G1 — No nonnegative comparison is imposed on negative prefix term a_1.

Error controls and negative controls

Direction

Use an upper reference for convergence and a lower reference for divergence.

Eventual quantifier

After specifying N prove nonnegative comparison for every later term.

• A divergent upper series forces a smaller series to diverge. → reject — 2^{-n}≤1/n for n≥1, but the former converges and the latter diverges.
• Finitely many comparisons suffice. → reject — Every n≥N is required.
• a_1=-1 invalidates eventual comparison. → reject — The prefix is separated; the entire tail after N=2 is nonnegative.

Result and limitations

The main series converges to 35/8; the positive lower bound c_n=1 forces the d series to diverge.

Verification and source binding

1 declared checks for this guide; only the listed expressions are covered.

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: fd28c54d3495eae6e7b299be008c76fd110252a14dcfe00ac96ae8711826fd0a
• original source SHA-256: fd28c54d3495eae6e7b299be008c76fd110252a14dcfe00ac96ae8711826fd0a
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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