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?
Sum
byTable
such as inTable[(-1)^(k + 1)/k FactorialPower[2, k], {k, 1, 1000}]
is rather convincing that the series converges... Presumably, MMA does not detect thatFactorialPower
should simplify. $\endgroup$ – anderstood Jan 4 '18 at 19:09factpow[x_, k_] := Product[(x - j), {j, 0, k - 1}]
,factpow[x, k]
simplifies to(1 - k + x) Pochhammer[2 - k + x, -1 + k]
, but still the sum does not converge. $\endgroup$ – anderstood Jan 4 '18 at 19:11NSum[(-1)^(k + 1)/k *FactorialPower[2, k], {k, 1, Infinity}]
returnsComplexInfinity
... $\endgroup$ – anderstood Jan 4 '18 at 19:13