---
tag: 0T32ME0A6H
kind: statement
role: theorem
name: Power series of arctangent
aliases: [gregory-series]
msc: [30B10, 33B10]
uses: [065XW2V6C3, 05P4B4Q22V, 01X5CVV1Q5, 0S4EMDQS2F, 037FD5NBRT]
sources:
  - url: https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
    locator: Infinite series
---

## Statement

For every real $x$ with $|x| \le 1$,
$$\arctan x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{2n + 1} = x - \frac{x^3}{3} + \frac{x^5}{5} - \cdots.$$

## Remarks

For $|x| < 1$, integrate the [[geometric series|01X5CVV1Q5]] $\frac{1}{1 + t^2} = \sum (-1)^n t^{2n}$ term by term from $0$ to $x$. Its sum is the [[derivative of arctangent|05P4B4Q22V]].
At $x = \pm 1$, the series converges by the alternating series test, and [[Abel's theorem|0S4EMDQS2F]] gives the equality. At $x = 1$ this is the [[Leibniz formula|037FD5NBRT]] $\pi/4 = 1 - 1/3 + 1/5 - \cdots$.
The radius of convergence is $1$. Mathlib states the series for $|x| < 1$ as `Real.hasSum_arctan`.

## Explanations

### L1 human

The arctangent is a power series with odd powers and the simple coefficients $\pm 1/(2n+1)$.
At $x = 1$, it gives a famous series for $\pi/4$.