Possible Duplicate:
The sum of $(-1)^n \frac{\ln n}{n}$
Compute $$\sum_{k=2}^{\infty}\frac{(-1)^{k}{\ln{k}}}{k}$$
Possible Duplicate:
The sum of $(-1)^n \frac{\ln n}{n}$
Compute $$\sum_{k=2}^{\infty}\frac{(-1)^{k}{\ln{k}}}{k}$$
Consider, for $s>1$, the following auxiliary convergent series: $$ \sum_{k=1}^\infty (-1)^k \frac{\log(k)}{k^s} = -\frac{\mathrm{d}}{\mathrm{d} s} \sum_{k=1}^\infty (-1)^k \frac{1}{k^s}= -\frac{\mathrm{d}}{\mathrm{d} s}\left( (2^{1-s} - 1)\zeta(s) \right) $$ The value of the series in question is obtained as a limit: $$ \sum_{k=1}^\infty (-1)^k \frac{\log(k)}{k} = \lim_{s \searrow 1} \left( 2^{1-s} \log(2) \zeta(s) + \zeta^\prime(s) (1-2^{1-s}) \right) $$ Since $\zeta(s) = \frac{1}{s-1} + \gamma + \mathcal{O}(s-1)$, and $\zeta^\prime(s) = -\frac{1}{(s-1)^2} + \mathcal{O}(1)$ we arrive at: $$ \sum_{k=1}^\infty (-1)^k \frac{\log(k)}{k} = \gamma \log(2) - \frac{\log^2(2)}{2} $$