Sum over k from 2 to Infinity of Zeta of k Over k Alternating in Sign/Proof 2
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Theorem
\(\ds \sum_{k \mathop = 2}^\infty \dfrac {\paren {-1}^k} k \map \zeta k\) | \(=\) | \(\ds \dfrac {\map \zeta 2} 2 - \dfrac {\map \zeta 3} 3 + \dfrac {\map \zeta 4} 4 - \dfrac {\map \zeta 5} 5 + \cdots\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \gamma\) |
Proof
\(\ds \map \ln {\map \Gamma {x + 1} }\) | \(=\) | \(\ds -\gamma x + \sum_{k \mathop = 2}^{\infty} \dfrac {\map \zeta k \paren {-x}^k} k\) | Power Series Expansion for Logarithm of Gamma Function | |||||||||||
\(\ds \) | \(=\) | \(\ds -\gamma x + \dfrac {\map \zeta 2 x^2} 2 - \dfrac {\map \zeta 3 x^3} 3 + \dfrac {\map \zeta 4 x^4} 4 - \dfrac {\map \zeta 5 x^5} 5 + \cdots\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \ln {\map \Gamma {1 + 1} }\) | \(=\) | \(\ds -\gamma + \sum_{k \mathop = 2}^{\infty} \dfrac {\map \zeta k \paren {-1}^k} k\) | setting $x = 1$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds 0\) | \(=\) | \(\ds -\gamma + \dfrac {\map \zeta 2} 2 - \dfrac {\map \zeta 3} 3 + \dfrac {\map \zeta 4} 4 - \dfrac {\map \zeta 5} 5 + \cdots\) | $\map \Gamma 2 = 1$ and Natural Logarithm of 1 is 0 | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \gamma\) | \(=\) | \(\ds \dfrac {\map \zeta 2} 2 - \dfrac {\map \zeta 3} 3 + \dfrac {\map \zeta 4} 4 - \dfrac {\map \zeta 5} 5 + \cdots\) | rearranging |
$\blacksquare$