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


Sources

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