I am often working with divergent series all around being this the bread and butter for a theoretical physicist. Thanks to the excellent work of Hardy these have lost their mystical Aurea and so, they have found some applications in concrete computations. In these days I have been involved with the following divergent series
$S=\sum_{n=1}^\infty\frac{2^n}{n}$
that is clearly divergent. Wolfram Alpha provides the following
$S_m=\sum_{n=1}^m\frac{2^n}{n}=-i\pi-2^{m+1}\Phi(2,1,m+1)$
being $\Phi(z,s,a)$ the Lerch function (see also Wikipedia). Is there any summation technique in this case?