somebody know the value of the following divergent serie
sum[(-1)^m e^m,{1,Infinity}]
What capea wrote is 'almost true'... in the sense I try to explain now...
The so called 'Dirichlet eta function' is defined as...
$\displaystyle \eta (s)= - \sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^{s}} = (1-2^{1-s})\ \zeta(s)$ (1)
Writing the (1) as...
$\displaystyle \eta (s)= \frac{1}{2} + \frac{1}{2}\ \sum_{n=1}^{\infty} (-1)^{n-1}\ \{n^{-s} - (n+1)^{-s}\}$ (2)
... You can, with a little of patience, demonstrate that (2) converges for $\displaystyle \text{Re}\ s > -1$ and that is...
$\displaystyle \eta^{'} (0)= \frac{1}{2}\ \ln \frac{\pi}{2}$ (3)
Now if You compute 'directly' the derivative of (1) You obtain...
$\displaystyle - \eta^{'} (s) = \sum_{n=1}^{\infty} \frac{(-1)^{n}\ \ln n}{n^{s}} $ (4)
... so that setting in (4) $\displaystyle s=0$ and considering (3) You obtain...
$\displaystyle \sum_{n=1}^{\infty} (-1)^{n} \ln n = - \frac{1}{2}\ \ln \frac{\pi}{2}$ (5)
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$