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

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

Let $n \in \Z_{> 0}$ be a (strictly) positive integer.

\(\ds E_{2 n}\) \(=\) \(\ds \paren {-1}^{n + 1} \dfrac {2^{2 n + 2} \paren {2 n}!} {\pi^{2 n + 1} } \sum_{j \mathop = 0}^\infty \frac {\paren {-1}^j} {\paren {2 j + 1}^{2 n + 1} }\)
\(\ds \) \(=\) \(\ds \paren {-1}^{n + 1} \dfrac {2^{2 n + 2} \paren {2 n}!} {\pi^{2 n + 1} } \paren {\frac 1 {1^{2 n + 1} } - \frac 1 {3^{2 n + 1} } + \frac 1 {5^{2 n + 1} } - \frac 1 {7^{2 n + 1} } + \cdots}\)

where $E_n$ is the $n$th Euler number.


Proof

\(\ds \paren {-1}^{n + 1} \dfrac {\pi^{2 n + 1} E_{2 n} } {2^{2 n + 2} \paren {2 n}!}\) \(=\) \(\ds \sum_{j \mathop = 0}^\infty \dfrac {\paren {-1}^j} {\paren {2 j + 1}^{2 n + 1} }\) Sum of Reciprocals of Odd Powers of Odd Integers Alternating in Sign
\(\ds \leadsto \ \ \) \(\ds \paren {-1}^{n + 1} E_{2 n}\) \(=\) \(\ds \dfrac {2^{2 n + 2} \paren {2 n}!} {\pi^{2 n + 1} } \sum_{j \mathop = 0}^\infty \frac {\paren {-1}^j} {\paren {2 j + 1}^{2 n + 1} }\) multiplying both sides by $\dfrac {2^{2 n + 2} \paren {2 n}!} {\pi^{2 n + 1} }$
\(\ds \leadsto \ \ \) \(\ds \paren {-1}^{2 n + 2} E_{2 n}\) \(=\) \(\ds \paren {-1}^{n + 1} \dfrac {2^{2 n + 2} \paren {2 n}!} {\pi^{2 n + 1} } \sum_{j \mathop = 0}^\infty \frac {\paren {-1}^j} {\paren {2 j + 1}^{2 n + 1} }\) multiplying both sides by $\paren {-1}^{n + 1}$
\(\ds \leadsto \ \ \) \(\ds E_{2 n}\) \(=\) \(\ds \paren {-1}^{n + 1} \dfrac {2^{2 n + 2} \paren {2 n}!} {\pi^{2 n + 1} } \sum_{j \mathop = 0}^\infty \frac {\paren {-1}^j} {\paren {2 j + 1}^{2 n + 1} }\) $\paren {-1}^{2 n + 2} = 1$ as $2 n + 2$ is even

$\blacksquare$


Sources

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