Sum of Reciprocals of Powers of Odd Integers Alternating in Sign/Corollary

Corollary to Sum of Reciprocals of Powers of Odd Integers Alternating in Sign

$\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {\paren {2 n + 1}^s} = \frac 1 {\map \Gamma s} \int_1^\infty \frac {\ln^{s - 1} x} {x^2 + 1} \rd x$

Proof

\(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {\paren {2 n + 1}^s}\)

\(=\)

\(\ds \frac 1 {2 \map \Gamma s} \int_0^\infty x^{s - 1} \map \sech x \rd x\)

Sum of Reciprocals of Powers of Odd Integers Alternating in Sign

\(\ds \)

\(=\)

\(\ds \frac 1 {\map \Gamma s} \int_0^\infty \frac {x^{s - 1} } {e^x + e^{- x} } \rd x\)

Definition of Hyperbolic Secant

\(\ds \)

\(=\)

\(\ds \frac 1 {\map \Gamma s} \int_0^\infty \frac {\paren {\ln e^x}^{s - 1} e^x} {\paren {e^x}^2 + 1} \rd x\)

\(\ds \)

\(=\)

\(\ds \frac 1 {\map \Gamma s} \int_1^\infty \frac 1 t \frac {\map {\ln^{s - 1} } t t} {t^2 + 1} \rd t\)

substituting $t = e^x$

\(\ds \)

\(=\)

\(\ds \frac 1 {\map \Gamma s} \int_1^\infty \frac {\ln^{s - 1} t} {t^2 + 1} \rd t\)

$\blacksquare$