Sum over k from 2 to Infinity of Zeta of k Over k Alternating in Sign/Proof 2

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 $

$=$

$\ds \gamma$

$\ds \map \ln {\map \Gamma {x + 1} }$

$=$

$\ds -\gamma x + \sum_{k \mathop = 2}^{\infty} \dfrac {\map \zeta k \paren {-x}^k} k$

$\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 \ \ $

$\ds 0$

$=$

$\ds -\gamma + \dfrac {\map \zeta 2} 2 - \dfrac {\map \zeta 3} 3 + \dfrac {\map \zeta 4} 4 - \dfrac {\map \zeta 5} 5 + \cdots$

$\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

$\blacksquare$