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

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

$\blacksquare$