Sum over k from 1 to Infinity of Zeta of 2k Over Odd Powers of 2/Proof 1
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Theorem
| \(\ds \sum_{k \mathop = 1}^\infty \dfrac {\map \zeta {2k} } {2^{2k - 1} }\) | \(=\) | \(\ds \dfrac {\map \zeta {2 } } 2 + \dfrac {\map \zeta {4 } } {2^3} + \dfrac {\map \zeta {6 } } {2^5} + \dfrac {\map \zeta {8 } } {2^7} + \cdots\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 1\) |
Proof
| \(\ds \map \zeta {2k}\) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \dfrac 1 {n^{2 k} }\) | Definition of Riemann Zeta Function | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \sum_{k \mathop = 1}^\infty \dfrac {\map \zeta {2k} } {2^{2k - 1} }\) | \(=\) | \(\ds \sum_{k \mathop = 1}^\infty \dfrac 2 {\paren 2^{2 k} } \sum_{n \mathop = 1}^\infty \dfrac 1 {n^{2 k} }\) | summing both sides as appropriate | ||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 1}^\infty \sum_{n \mathop = 1}^\infty \dfrac 2 {\paren {2}^{2 k} \paren n^{2 k} }\) | Tonelli's Theorem: Corollary | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \sum_{k \mathop = 1}^\infty \sum_{n \mathop = 1}^\infty \dfrac 1 {\paren {4 n^2}^k }\) | moving the $2$ outside | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \sum_{n \mathop = 1}^\infty \dfrac {\dfrac 1 {\paren {4 n^2} } } {\paren {1 - \dfrac 1 {\paren {4 n^2} } } }\) | Sum of Infinite Geometric Sequence: Corollary $1$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \sum_{n \mathop = 1}^\infty \dfrac {\dfrac 1 {\paren {4 n^2} } } {\paren {1 - \dfrac 1 {\paren {4 n^2} } } } \times \dfrac {4 n^2} {4 n^2}\) | multiplying by $1$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \sum_{n \mathop = 1}^\infty \dfrac 1 {4 n^2 - 1 }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \sum_{n \mathop = 1}^\infty \dfrac 1 2 \paren {\dfrac 1 {2 n - 1 } - \dfrac 1 {2 n + 1 } }\) | Definition of Partial Fractions Expansion | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \paren {\dfrac 1 {2 n - 1 } - \dfrac 1 {2 n + 1 } }\) | Definition of Telescoping Series | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {1 - \dfrac 1 3} + \paren {\dfrac 1 3 - \dfrac 1 5} + \paren {\dfrac 1 5 - \dfrac 1 7} + \paren {\dfrac 1 7 - \dfrac 1 9} + \cdots\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 1\) |
Hence:
| \(\ds \sum_{k \mathop = 1}^\infty \dfrac {\map \zeta {2k} } {2^{2k - 1} }\) | \(=\) | \(\ds 1\) |
$\blacksquare$