Gregory Series

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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 Gregory series.


Proof

\(\ds 1\) \(=\) \(\ds \frac {\sec^2 \theta} {\sec^2 \theta}\)
\(\ds \) \(=\) \(\ds \sec^2 \theta \times \frac 1 {1 - \paren {-\tan^2 \theta} }\)
\(\ds \) \(=\) \(\ds \sec^2 \theta \times \sum_{n \mathop = 0}^\infty \paren {-\tan^2 \theta}^n\) Sum of Infinite Geometric Sequence
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \sec^2 \theta \tan^{2 n} \theta\)


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
\(\ds \int \rd \theta\) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \paren {-1} ^ n \int \paren { \sec ^ 2 \theta \tan ^ {2 n} \theta} \rd \theta\)
\(\ds \theta\) \(=\) \(\ds \sum_{n \mathop = 0} ^ \infty \frac {\paren {-1} ^ n}{2 n + 1} \tan ^ {2 n + 1} \theta\) Primitive of Power of Tangent of a x by Square of Secant of a x

$\blacksquare$


Also presented as

This series is also presented as:

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

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


Also known as

This is also known as Gregory's series.


Also see


Source of Name

This entry was named for James Gregory.


Historical Note

James Gregory established the result now known as the Gregory Series in $1671$, or perhaps earlier.


Sources

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