Gregory Series

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Theorem

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

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


Proof

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


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


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

$\blacksquare$


Also 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.