Sum over k from 1 to Infinity of Zeta of 2k Over Odd Powers of 2/Proof 3
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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
From Laurent Series Expansion for Cotangent Function, we have:
- $\ds \pi \cot \pi z = \dfrac 1 z - 2 \sum_{k \mathop = 1}^\infty \map \zeta {2 k} z^{2 k - 1}$
Setting $z = \dfrac 1 2$:
\(\ds \pi \map \cot {\dfrac \pi 2}\) | \(=\) | \(\ds 2 - 2 \sum_{k \mathop = 1}^\infty \dfrac {\map \zeta {2k} } {2^{2k - 1} }\) | Laurent Series Expansion for Cotangent Function setting $z = \dfrac 1 2$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 0\) | \(=\) | \(\ds 2 - 2 \sum_{k \mathop = 1}^\infty \dfrac {\map \zeta {2k} } {2^{2k - 1} }\) | Cotangent of Right Angle | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds 2 \sum_{k \mathop = 1}^\infty \dfrac {\map \zeta {2k} } {2^{2k - 1} }\) | \(=\) | \(\ds 2\) | |||||||||||
\(\ds \) | \(=\) | \(\ds 1\) | dividing both sides by $2$ |
Hence:
\(\ds \sum_{k \mathop = 1}^\infty \dfrac {\map \zeta {2k} } {2^{2k - 1} }\) | \(=\) | \(\ds 1\) |
$\blacksquare$