Sum over k from 1 to Infinity of Zeta of 2k Over Odd Powers of 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 1
\(\ds \map \zeta {2k}\) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \dfrac 1 {n^{2 k} }\) | 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:
\(\ds \sum_{k \mathop = 1}^\infty \dfrac {\map \zeta {2k} } {2^{2k - 1} }\) | \(=\) | \(\ds 1\) |
$\blacksquare$
Proof 2
\(\ds \map \zeta {2k}\) | \(=\) | \(\ds \frac 1 {\map \Gamma {2 k} } \int_0^\infty \frac {t^{2 k - 1} } {e^t - 1} \rd t\) | 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:
\(\ds \sum_{k \mathop = 1}^\infty \dfrac {\map \zeta {2k} } {2^{2k - 1} }\) | \(=\) | \(\ds 1\) |
$\blacksquare$
Proof 3
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:
\(\ds \sum_{k \mathop = 1}^\infty \dfrac {\map \zeta {2k} } {2^{2k - 1} }\) | \(=\) | \(\ds 1\) |
$\blacksquare$
Also see
- Sum over k from 1 to Infinity of Zeta of 2k Minus One
- Sum over k from 2 to Infinity of Zeta of k Minus One
Sources
- robjohn (https://math.stackexchange.com/users/44121/robjohn), How to prove that $\dfrac {\zeta 2} 2 + \dfrac {\zeta 4} {2^3} + \cdots$, URL (version: 2014-12-24): https://math.stackexchange.com/q/1080000