# Infinite series

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

An infinite series is defined through the limit of its partial-sum sequence; terms tending to zero are necessary but not sufficient for convergence.

mko-infinite-series · definition · v0.10.0



## Mathematical statement

This slice handles exact geometric and telescoping series. Retain the starting-index convention, partial sum and remainder. The geometric sum formula is used only for |r|<1. A long finite prefix or terms tending to zero cannot alone establish convergence.





## Representative formula (exact TeX)

S(a)





## Assumptions


- Terms and partial sums use exact rational arithmetic.
- The symbolic tail explicitly covers every applicable n.





## Symbol scope


- S — series record
- a — term sequence





## Intuition, definition, and misconceptions

Partial sums, not individual terms, define the series limit.

Retain |r|<1 for the geometric formula.

Keep telescoping remainders; terms tending to zero and finite prefixes alone are insufficient.

Neither terms tending to zero nor an apparently stable long finite prefix proves a series convergent.





## Adopted derivation guide

[Complete tails for geometric and telescoping series](https://ocme.evemisslab.com/en/derivations/derivation-mko-infinite-series-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Sequence](https://ocme.evemisslab.com/en/mko/mko-sequence/) — Terms and partial sums are sequences.
- [Sequence limits and exact tail cutoffs](https://ocme.evemisslab.com/en/mko/mko-sequence-limit/) — Series convergence is a partial-sum limit.
- [Arithmetic operations](https://ocme.evemisslab.com/en/mko/mko-arithmetic-operations/) — Finite sums use exact operations.





## Verification boundary

Only the explicit symbolic geometric and telescoping families are checked.

not_formalized

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





## Next obligation

Add absolute convergence 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/infinite-series.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-infinite-series/index.md)