# Series & Convergence

**Type:** Concept  
**Domain:** Analysis  
**Codex URL:** /mathematics/analysis/series-and-convergence/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
Infinite sums that may converge to a finite value or diverge. Resolves Zeno's Paradoxes; basis of Taylor series and the Basel Problem.

## Geometric series
Σarⁿ = a/(1−r), converges iff |r|<1

## Harmonic series diverges
1+½+⅓+¼+... diverges (Oresme, 14th century) despite terms shrinking to 0.

## Convergence tests
Ratio test, comparison test, alternating series test, integral test.

## Absolute vs conditional convergence
Riemann Rearrangement Theorem: conditionally convergent series can be reordered to sum to ANY value.

## The Basel Problem
Euler (1734): 1+1/4+1/9+1/16+... = π²/6. Generalizes to Riemann zeta function ζ(s)=Σ1/nˢ.

## Sources
### Tier 1
- Rudin, W. (1976). *Principles of Mathematical Analysis*. 3rd ed.
- Knopp, K. (1990 reprint). *Theory and Application of Infinite Series*.
### Tier 2
- Hardy, G.H. (1949/1991). *Divergent Series*.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*