Sum over k from 1 to Infinity of Zeta of 2k Over Odd Powers of 2/Proof 2
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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
| \(\ds \map \zeta {2k}\) | \(=\) | \(\ds \frac 1 {\map \Gamma {2 k} } \int_0^\infty \frac {t^{2 k - 1} } {e^t - 1} \rd t\) | Integral Representation of Riemann Zeta Function in terms of Gamma Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {\paren {2 k - 1}! } \int_0^\infty \frac {t^{2 k - 1} } {e^t - 1} \rd t\) | Gamma Function Extends Factorial | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \sum_{k \mathop = 1}^\infty \frac {\map \zeta {2k} } {2^{2k - 1} }\) | \(=\) | \(\ds \sum_{k \mathop = 1}^\infty \int_0^\infty \frac {t^{2 k - 1} } {2^{2k - 1} \paren {2 k - 1}!} \frac 1 {e^t - 1} \rd t\) | summing both sides as appropriate | ||||||||||
| \(\ds \) | \(=\) | \(\ds \int_0^\infty \sum_{k \mathop = 1}^\infty \frac {\paren {\dfrac t 2}^{2 k - 1} } {\paren {2 k - 1}!} \dfrac 1 {e^t - 1} \rd t\) | Tonelli's Theorem | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int_0^\infty \frac {\map \sinh {\dfrac t 2} } {e^t - 1} \rd t\) | Power Series Expansion for Hyperbolic Sine Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int_0^\infty \frac {e^{\frac t 2} - e^{-\frac t 2} } {2 \paren {e^t - 1} } \rd t\) | Definition of Hyperbolic Sine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 2 \int_0^\infty \frac {e^{-\frac t 2} \paren {e^t - 1 } } {\paren {e^t - 1 } } \rd t\) | factoring out $e^{-\frac t 2}$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 2 \int_0^\infty e^{-\frac t 2} \rd t\) | canceling $\paren {e^t - 1 }$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 2 \bigintlimits {-2 e^{-\frac t 2} } 0 \infty\) | Primitive of Exponential Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds 1\) |
Hence:
| \(\ds \sum_{k \mathop = 1}^\infty \dfrac {\map \zeta {2k} } {2^{2k - 1} }\) | \(=\) | \(\ds 1\) |
$\blacksquare$