---
tag: 02KTRHQMHN
kind: statement
role: theorem
name: Power series of sine
msc: [33B10, 30B10]
uses: [0WB57D9488, 0JBMYAA9QH, 043SJ5RF59]
sources:
  - url: https://en.wikipedia.org/wiki/Sine_and_cosine
    locator: Series and polynomials
---

## Statement

For every real $x$, the series $\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$ converges absolutely, and
$$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots.$$
This series is the [[Maclaurin series|043SJ5RF59]] of the sine function.

## Remarks

The [[ratio of the terms|0JBMYAA9QH]] is $\frac{x^2}{(2n+2)(2n+3)} \to 0$. So the radius of convergence is $\infty$.
With the series as the definition of $\sin$, the equality holds by definition. With a geometric definition, Taylor's theorem proves it.
For $|x| \le 1$, the error after the term $x^{2n+1}$ is at most $\frac{1}{(2n+3)!}$. Mathlib states the series as `Real.hasSum_sin`.

## Explanations

### L1 human

Sine is an infinite odd polynomial with alternating signs.
Near $0$, $\sin x$ is close to $x$, and closer still to $x - x^3/6$.