Infinite series

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

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Sequence — Terms and partial sums are sequences.
• Sequence limits and exact tail cutoffs — Series convergence is a partial-sum limit.
• 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.