Sum of Reciprocals of Even Powers of Odd Integers/Corollary

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Corollary to Sum of Reciprocals of Even Powers of Odd Integers

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

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


Proof

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

$\blacksquare$


Sources

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