Sum of Reciprocals of Sixth Powers Alternating in Sign

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Theorem

\(\ds \sum_{n \mathop = 1}^\infty \dfrac {\paren {-1}^{n + 1} } {n^6}\) \(=\) \(\ds \frac 1 {1^6} - \frac 1 {2^6} + \frac 1 {3^6} - \frac 1 {4^6} + \cdots\)
\(\ds \) \(=\) \(\ds \frac {31 \pi^6} {30 \, 240}\)


Proof

\(\ds \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } {n^6}\) \(=\) \(\ds \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n - 1}^6} - \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n}^6}\) separating odd positive terms from even negative terms
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n - 1}^6} - \frac 1 {64} \sum_{n \mathop = 1}^\infty \frac 1 {n^6}\)
\(\ds \) \(=\) \(\ds \frac {\pi^6} {960} - \frac 1 {64} \sum_{n \mathop = 1}^\infty \frac 1 {n^6}\) Sum of Reciprocals of Sixth Powers of Odd Integers
\(\ds \) \(=\) \(\ds \frac {\pi^6} {960} - \frac 1 {64} \times \frac {\pi^6} {945}\) Riemann Zeta Function of 6
\(\ds \) \(=\) \(\ds \frac {\pi^6 \paren {63 - 1} } {60 \, 480}\)
\(\ds \) \(=\) \(\ds \frac {31 \pi^6} {30 \, 240}\)

$\blacksquare$


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