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 | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\pi^6 \paren {63 - 1} } {60 \, 480}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {31 \pi^6} {30 \, 240}\) |
$\blacksquare$
Also see
- Sum of Reciprocals of Even Powers of Integers Alternating in Sign
- Sum of Reciprocals of Squares Alternating in Sign
- Sum of Reciprocals of Fourth Powers Alternating in Sign
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 19$: Series involving Reciprocals of Powers of Positive Integers: $19.24$