Sum of Reciprocals of Fourth Powers Alternating in Sign

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Theorem

\(\ds \sum_{n \mathop = 1}^\infty \dfrac {\paren {-1}^{n + 1} } {n^4}\) \(=\) \(\ds \frac 1 {1^4} - \frac 1 {2^4} + \frac 1 {3^4} - \frac 1 {4^4} + \cdots\)
\(\ds \) \(=\) \(\ds \dfrac {7 \pi^4} {720}\)


Proof

\(\ds \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } {n^4}\) \(=\) \(\ds \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n - 1}^4} - \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n}^4}\) separating odd positive terms from even negative terms
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n - 1}^4} - \frac 1 {16} \sum_{n \mathop = 1}^\infty \frac 1 {n^4}\)
\(\ds \) \(=\) \(\ds \frac {\pi^4} {96} - \frac 1 {16} \sum_{n \mathop = 1}^\infty \frac 1 {n^4}\) Sum of Reciprocals of Fourth Powers of Odd Integers
\(\ds \) \(=\) \(\ds \frac {\pi^4} {96} - \frac 1 {16} \times \frac {\pi^4} {90}\) Riemann Zeta Function of 4
\(\ds \) \(=\) \(\ds \frac {\pi^4 \paren {15 - 1} } {1440}\)
\(\ds \) \(=\) \(\ds \frac {7 \pi^4} {720}\)

$\blacksquare$


Also see


Sources

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