Sum over k from 2 to Infinity of Zeta of k Over Powers of 2 Alternating in Sign
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Theorem
| \(\ds \sum_{k \mathop = 2}^\infty \dfrac {\paren {-1}^k \map \zeta k} {2^k}\) | \(=\) | \(\ds \dfrac {\map \zeta 2} {2^2} - \dfrac {\map \zeta 3} {2^3} + \dfrac {\map \zeta 4 } {2^4} - \dfrac {\map \zeta 5} {2^5} + \cdots\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 1 - \ln 2\) |
Proof
| \(\ds \sum_{k \mathop = 2}^\infty \dfrac {\paren {-1}^k \map \zeta k } {2^k}\) | \(=\) | \(\ds \sum_{k \mathop = 1}^\infty \dfrac {\paren {-1}^{k + 1} \map \zeta {k + 1} } {2^{k + 1} }\) | $k \to k + 1$ and reindexing the sum | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 1}^\infty \sum_{n \mathop = 1}^\infty \paren {\dfrac {-1} {2 n} }^{k + 1}\) | Definition of Riemann Zeta Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \sum_{k \mathop = 1}^\infty \paren {\dfrac {-1} {2 n} }^{k + 1}\) | Fubini's Theorem | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \dfrac {-1} {2 n} \sum_{k \mathop = 1}^\infty \paren {\dfrac {-1} {2 n} }^k\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \paren {\dfrac {-1} {2 n} } \times \dfrac {\paren {\dfrac {-1} {2 n} } } {1 - \paren {\dfrac {-1} {2 n} } }\) | Sum of Infinite Geometric Sequence: Corollary $1$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \dfrac 1 {2 n \paren {2 n + 1} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \paren {\dfrac 1 {2 n} - \dfrac 1 {\paren {2 n + 1} } }\) | Partial Fraction Decomposition | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 2 - \frac 1 3 + \frac 1 4 - \frac 1 5 + \cdots\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 1 - \paren {1 - \frac 1 2 + \frac 1 3 - \frac 1 4 + \dotsb }\) | adding $0$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds 1 - \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^\paren {n - 1} } n\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 1 - \ln 2\) | Definition of Mercator's Constant |
$\blacksquare$