Absolute convergence of series

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

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

Prerequisites

• Infinite series — Convergence concerns partial sums.
• Absolute value — The absolute series takes the absolute value of each term.
• Sequence limits and exact tail cutoffs — 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.