Power Series Expansion for Secant Function

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Theorem

The (real) secant function has a Taylor series expansion:

\(\ds \sec x\) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {E_{2 n} x^{2 n} } {\paren {2 n}!}\)
\(\ds \) \(=\) \(\ds 1 + \frac {x^2} 2 + \frac {5 x^4} {24} + \frac {61 x^6} {720} + \dfrac {1385 x^8} {40320} + \cdots\)


where $E_{2 n}$ denotes the Euler numbers.

This converges for $\size x < \dfrac \pi 2$.


Proof

\(\ds \sec x\) \(=\) \(\ds \map \sech {i x}\) Secant in terms of Hyperbolic Secant
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \frac {E_n \paren {i x}^n} {n!}\) Definition of Euler Numbers
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \frac {E_{2 n} \paren {i x}^{2 n} } {\paren {2 n}!}\) Odd terms vanish
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n E_{2 n} x^{2 n} } {\paren {2 n}!}\)

$\blacksquare$


Also presented as

The Power Series Expansion for Secant Function can also be presented in the form:

\(\ds \sec x\) \(=\) \(\ds 1 + \sum_{n \mathop = 1}^\infty \frac { {E_n}^* x^{2 n} } {\paren {2 n}!}\)

where ${E_n}^*$ denotes the alternative form of the Euler numbers.


Sources

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