# Absolute convergence of series

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

A series is absolutely convergent when the sum of absolute terms converges; absolute convergence implies convergence, but the converse does not hold in general.

mko-absolute-convergence-of-series · definition · v0.10.0



## Mathematical statement

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.





## Representative formula (exact TeX)

A(S)





## Assumptions


- The index convention and exact rational partial sums and tails are fixed.
- A conditional-convergence example has both convergence and absolute-divergence certificates.





## Symbol scope


- A — absolute-series convergence
- S — series





## Intuition, definition, and misconceptions

Check the sum of absolute terms, not the absolute value of the signed sum.

Absolute convergence implies convergence.

Alternation alone is insufficient; conditional convergence needs both certificates.

Terms tending to zero, boundedness or alternation alone do not establish absolute convergence.





## Adopted derivation guide

[Absolute convergence and the two certificates for alternating harmonic](https://ocme.evemisslab.com/en/derivations/derivation-mko-absolute-convergence-of-series-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Infinite series](https://ocme.evemisslab.com/en/mko/mko-infinite-series/) — Convergence concerns partial sums.
- [Absolute value](https://ocme.evemisslab.com/en/mko/mko-absolute-value/) — The absolute series takes the absolute value of each term.
- [Sequence limits and exact tail cutoffs](https://ocme.evemisslab.com/en/mko/mko-sequence-limit/) — The two series have distinct convergence obligations.





## Verification boundary

Only the explicit geometric and alternating-harmonic families are handled.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add comparison tests and power series.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/absolute-convergence-of-series.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-absolute-convergence-of-series/index.md)