Absolute convergence and the two certificates for alternating harmonic

Goal and scope

Distinguish absolute convergence of a negative-ratio geometric series from conditional convergence of alternating harmonic.

Exact geometric tails and harmonic blocks; conditional convergence needs both signed convergence and absolute divergence.

Givens and symbols

• algebra
• epsilon
• monotone_complete
• definition_here

• G1 — a_n=(-1/2)^n for n≥0; b_n=(-1)^{n+1}/n for n≥1.

• B_N — alternating-harmonic partial sum
• B — proved existing limit

Rules and sources

• mko_definition · cited — Geometric series use |r|<1 for absolute convergence. The alternating harmonic example combines an alternating-remainder certificate of convergence with harmonic divergence of the absolute series. Bounded terms, terms tending to zero, or the absolute value of a partial sum do not establish absolute convergence. · mko-absolute-convergence-of-series
• 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.
• monotone_complete · prerequisite — Every bounded monotone real sequence converges; partial sums of a nonnegative series are nondecreasing.
• definition_here · prerequisite — Absolute convergence means convergence of the termwise-absolute-value series; conditional convergence means signed convergence without absolute convergence.

Main derivation

|a_n|=2^{-n}

\sum_{n=0}^{N}|a_n|=2-2^{-N}

2^{-N}\le1/(N+1)\to0

\sum_{n=0}^{\infty}|a_n|=2

B_{2m}=\sum_{n=1}^{2m}b_n

B_{2(m+1)}-B_{2m}=1/(2m+1)-1/(2m+2)>0

B_{2m+1}=B_{2m}+1/(2m+1)

0\le B_{2m}<B_{2m+1}\le1

B_{2m}\uparrow,\ B_{2m+1}\downarrow,\ B_{2m+1}-B_{2m}=1/(2m+1)\to0

B_N\to B,\quad |B-B_N|\le1/(N+1)

\sum_{n=2^{j-1}+1}^{2^j}|b_n|

\sum_{n=2^{j-1}+1}^{2^j}1/n\ge2^{j-1}/2^j=1/2

\sum_{n=1}^{2^m}|b_n|\ge1+m/2

\sum_{n=1}^{\infty}|b_n|\text{ diverges}

\sum a_n\text{ absolutely convergent};\quad\sum b_n\text{ convergent but not absolutely}

\sum b_n\text{ conditionally convergent}

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · S02, S05, S07 — Bounded terms, vanishing terms and alternation alone have not been promoted to certificates.
• S07 · premise_audit · S06 — Bounded signed partial sums do not imply bounded absolute partial sums.
• S06 · premise_audit · G1 — The lower bound holds for every integer j≥1, not observed blocks only.
• S05 · premise_audit · S03, S04 — The bound follows from bracketing, without substituting an unknown closed form for a proof.
• S04 · premise_audit · G1, S03 — Alternation is supported by decreasing 1/n magnitudes, not arbitrary signs.
• S03 · premise_audit · G1 — Use m≥1 or initialize B_0=0 consistently.
• S02 · premise_audit · S01 — Absolute convergence is not inferred merely from alternating signs.
• S01 · inverse_equivalence · G1 — Take absolute values termwise, not only after summing signed terms.

Error controls and negative controls

Two certificates

Conditional convergence needs signed convergence and absolute divergence.

Whole tails, not signs

Use full monotonicity and block formulas for every term.

• Alternation guarantees absolute convergence. → reject — The series of termwise absolute values of the alternating harmonic series was proved divergent.
• |sum b_n| equals sum |b_n|. → reject — Already |1-1/2|=1/2 while 1+1/2=3/2.
• Terms tending to zero suffice for absolute convergence. → reject — 1/n tends to zero while block lower bounds force divergence.

Result and limitations

The (-1/2)^n series is absolutely convergent; alternating harmonic is conditionally convergent.

Verification and source binding

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• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
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