I wanted to compute the series defined by
$ $ \sum_{k=1}^\infty\frac{(-1)^{k+1}}k x^\underline k$ $
where $ x^\underline k:=\prod_{j=0}^{k-1}(x-j)$ is a falling factorial. Thus I write
Sum[(-1)^(k + 1)/k FactorialPower[x, k], {k, 1, Infinity}] but Mathematica says (without more explanation) that “the series does not converges”. Thus I thought that this can be certainly true for some values of $ x$ so I tried to compute this time
Sum[(-1)^(k + 1)/k FactorialPower[2, k], {k, 1, Infinity}] what is a finite sum because FactorialPower[2,k] is zero when $ k>2$ . However Mathematica also said that “the series doe not converges”. So, what is going on? This is a bug or there is some technicality that Im not seeing?
