Sum of Reciprocals of Even Powers of Odd Integers
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Theorem
Let $n \in \Z_{> 0}$ be a (strictly) positive integer.
\(\ds \sum_{j \mathop = 1}^\infty \frac 1 {\paren {2 j - 1}^{2 n} }\) | \(=\) | \(\ds \dfrac 1 {1^{2 n} } + \dfrac 1 {3^{2 n} } + \dfrac 1 {5^{2 n} } + \dfrac 1 {7^{2 n} } + \cdots\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-1}^{n + 1} \dfrac {B_{2 n} \paren {2^{2 n} - 1} \pi^{2 n} } {2 \paren {2 n}!}\) |
Corollary
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
\(\ds \sum_{j \mathop = 1}^\infty \frac 1 {j^{2 n} }\) | \(=\) | \(\ds \sum_{j \mathop = 1}^\infty \frac 1 {\paren {2 j}^{2 n} } + \sum_{j \mathop = 1}^\infty \frac 1 {\paren {2 j - 1}^{2 n} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2^{2 n} } \sum_{j \mathop = 1}^\infty \frac 1 {j^{2 n} } + \sum_{j \mathop = 1}^\infty \frac 1 {\paren {2 j - 1}^{2 n} }\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {-1}^{n + 1} \dfrac {B_{2 n} 2^{2 n - 1} \pi^{2 n} } {\paren {2 n}!}\) | \(=\) | \(\ds \frac 1 {2^{2 n} } \times \paren {-1}^{n + 1} \dfrac {B_{2 n} 2^{2 n - 1} \pi^{2 n} } {\paren {2 n}!} + \sum_{j \mathop = 1}^\infty \frac 1 {\paren {2 j - 1}^{2 n} }\) | Riemann Zeta Function at Even Integers | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sum_{j \mathop = 1}^\infty \frac 1 {\paren {2 j - 1}^{2 n} }\) | \(=\) | \(\ds \paren {-1}^{n + 1} \dfrac {B_{2 n} 2^{2 n - 1} \pi^{2 n} } {\paren {2 n}!} - \frac 1 {2^{2 n} } \times \paren {-1}^{n + 1} \dfrac {B_{2 n} 2^{2 n - 1} \pi^{2 n} } {\paren {2 n}!}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-1}^{n + 1} \dfrac {B_{2 n} 2^{2 n} \pi^{2 n} } {2 \paren {2 n}!} - \paren {-1}^{n + 1} \dfrac {B_{2 n} \pi^{2 n} } {2 \paren {2 n}!}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-1}^{n + 1} \dfrac {B_{2 n} \paren {2^{2 n} - 1} \pi^{2 n} } {2 \paren {2 n}!}\) |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 19$: Series involving Reciprocals of Powers of Positive Integers: $19.36$