Gregory Series

Theorem

For $-\dfrac \pi 4 \le \theta \le \dfrac \pi 4$:

$\theta = \tan \theta - \dfrac 1 3 \tan^3 \theta + \dfrac 1 5 \tan^5 \theta - \ldots$

This is called the .

$\ds 1$

$=$

$\ds \frac {\sec^2 \theta} {\sec^2 \theta}$

$\ds $

$=$

$\ds \sec^2 \theta \times \frac 1 {1 - \paren {-\tan^2 \theta} }$

$\ds $

$=$

$\ds \sec^2 \theta \times \sum_{n \mathop = 0}^\infty \paren {-\tan^2 \theta}^n$

$\ds $

$=$

$\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \sec^2 \theta \tan^{2 n} \theta$

By the root test the radius of convergence is $-\dfrac \pi 4 \le \theta \le \dfrac \pi 4$.

$\ds \int \paren {1} \rd \theta$

$=$

$\ds \int \paren {\sum_{n \mathop = 0} ^ \infty \paren {-1} ^ n \sec ^ 2 \theta \tan ^ {2 n} \theta} \rd \theta$

Integrating both sides

$\ds \int \rd \theta$

$=$

$\ds \sum_{n \mathop = 0}^\infty \paren {-1} ^ n \int \paren { \sec ^ 2 \theta \tan ^ {2 n} \theta} \rd \theta$

$\ds \theta$

$=$

$\ds \sum_{n \mathop = 0} ^ \infty \frac {\paren {-1} ^ n}{2 n + 1} \tan ^ {2 n + 1} \theta$

$\blacksquare$

The can also be seen in the form:

$\tan^{-1} \theta = \theta - \dfrac {\theta^3} 3 + \dfrac {\theta^5} 5 - \ldots$

which is valid for $-1 \le x \le 1$.

The is also known as Gregory's series.

It can also be seen referred to as the inverse tangent series when expressed in the inverse tangent form.

This entry was named for James Gregory.

James Gregory established the result now known as the at least as early as $1671$.

Some sources suggest $1667$.

1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Gregory's series (J. Gregory, 1667)
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Gregory's series (J. Gregory, 1667)