Sum of Reciprocals of Odd Powers of Odd Integers Alternating in Sign
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It has been suggested that this page or section be merged into Dirichlet Beta Function at Odd Positive Integers. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Mergeto}} from the code. |
Theorem
Let $n \in \Z_{\ge 0}$ be a positive integer.
Then:
\(\ds \sum_{j \mathop = 0}^\infty \frac {\paren {-1}^j} {\paren {2 j + 1}^{2 n + 1} }\) | \(=\) | \(\ds \paren {-1}^{n + 1} \frac {\pi^{2 n + 1} E_{2 n} } {2^{2 n + 2} \paren {2 n}!}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {1^{2 n + 1} } - \frac 1 {3^{2 n + 1} } + \frac 1 {5^{2 n + 1} } - \frac 1 {7^{2 n + 1} } + \cdots\) |
where $E_n$ is the $n$th Euler number.
Corollary
Let $n \in \Z_{> 0}$ be a (strictly) positive integer.
\(\ds E_{2 n}\) | \(=\) | \(\ds \paren {-1}^{n + 1} \dfrac {2^{2 n + 2} \paren {2 n}!} {\pi^{2 n + 1} } \sum_{j \mathop = 0}^\infty \frac {\paren {-1}^j} {\paren {2 j + 1}^{2 n + 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-1}^{n + 1} \dfrac {2^{2 n + 2} \paren {2 n}!} {\pi^{2 n + 1} } \paren {\frac 1 {1^{2 n + 1} } - \frac 1 {3^{2 n + 1} } + \frac 1 {5^{2 n + 1} } - \frac 1 {7^{2 n + 1} } + \cdots}\) |
Proof
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Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 19$: Series involving Reciprocals of Powers of Positive Integers: $19.38$