Sum of Reciprocals of Powers of Odd Integers Alternating in Sign
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Theorem
- $\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {\paren {2 n + 1}^s} = \frac 1 {2 \map \Gamma s} \int_0^\infty x^{s - 1} \map \sech x \rd x$
where:
- $\map \Re s > 0$
- $\Gamma$ is the gamma function
- $\sech$ is the hyperbolic secant function.
Corollary
- $\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {\paren {2 n + 1}^s} = \frac 1 {\map \Gamma s} \int_1^\infty \frac {\ln^{s - 1} x} {x^2 + 1} \rd x$
Proof
\(\ds \int_0^\infty x^{s - 1} \map \sech x \rd x\) | \(=\) | \(\ds 2 \int_0^\infty \frac { x^{s - 1} } {e^x + e^{-x} } \rd x\) | Definition of Hyperbolic Secant | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \int_0^\infty \frac {x^{s - 1} e^{-x} } {1 - \paren {- e^{-2 x} } } \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 \int_0^\infty x^{s - 1} e^{-x} \sum_{n \mathop = 0}^\infty \paren {-1}^n e^{-2 n x} \rd x\) | Sum of Infinite Geometric Sequence | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \sum_{n \mathop = 0}^\infty \paren {-1}^n \int_0^\infty x^{s - 1} e^{-\paren {2 n + 1} x} \rd x\) | Fubini's Theorem | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \sum_{n \mathop = 0}^\infty \paren {-1}^n \int_0^\infty \paren {\frac t {2 n + 1} }^{s - 1} e^{-t} \frac {\rd t} {2 n + 1}\) | substituting $t = \paren {2 n + 1} x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {\paren {2 n + 1}^s} \int_0^\infty t^{s - 1} e^{-t} \rd t\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 \map \Gamma s \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {\paren {2 n + 1}^s}\) | Definition of Gamma Function |
So:
- $\ds \frac 1 {2 \map \Gamma s} \int_0^\infty x^{s - 1} \map \sech x \rd x = \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {\paren {2 n + 1}^s}$
$\blacksquare$