# Taylor Series

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

## Summary
Represents smooth functions as infinite polynomial sums built from derivatives.

## Formula
f(x) = f(a) + f'(a)(x−a) + f''(a)(x−a)²/2! + ...

## Maclaurin series (a=0)
eˣ = 1+x+x²/2!+...; sin(x) = x−x³/3!+...; cos(x) = 1−x²/2!+...

## Radius of convergence
Series may only converge within a certain radius around a — e.g. 1/(1−x) has radius 1 despite being defined more broadly.

## Analytic functions and a counterexample
e^(−1/x²) is infinitely differentiable at 0, all derivatives there are 0, yet the function isn't identically zero — its Taylor series fails to represent it.

## Lagrange remainder (1797)
Precise error bound for truncated Taylor polynomials.

## Historical origins
Taylor (1715); precursors Gregory, Newton; Mādhava of Sangamagrama (c.1400 CE, India) discovered equivalent series for sin/cos/arctan roughly 3 centuries earlier.

## Complex analysis contrast
Holomorphic (complex-differentiable) functions are automatically analytic — stronger than the real case.

## Sources
### Tier 1
- Taylor, B. (1715). Methodus Incrementorum.
- Rudin, W. (1976). Principles of Mathematical Analysis.
### Tier 2
- Roy, R. (2021). Series and Products in the Development of Mathematics.

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