Trouble showing this series converges

Question

I'm attempting to use Mathematica to verify that the following series converges:

$$\sum_{n=0}^\infty \sin(e\pi n!)$$

I tried

Sum[Sin[Exp[1]*Pi*Factorial[n]], {n, 0, Infinity}]

but Mathematica tells me that the series does not converge, even though it does converge. What am I doing wrong?

Answer 1

With your attempt you ask Mathematica for the exact value, but there is little hope to get it for this series, I guess. I played with the Regularization but with little success (at least it won't claim that it doesn't converge anymore). However, as you linked a post where it's already proven mathematically I assume that you are looking for a way to visualize that it converges. You can plot the series like this:

S[0] = 0;
S[k_] := S[k] = S[k - 1] + Block[{$MaxExtraPrecision = 10000}, N[Sin[π E k!], 10]];
pd = Table[{k, 1. S[k]}, {k, 0, 2000}];
ListPlot[pd, PlotRange -> {0, 1}]

That's a really nice series, by the way! ;-)

Answer 2

Just a small comment. The starting value of the series at k=0 is Sin[π E], which changes the result of the previous post (I have increased the final precision somewhat):

S[0] = Sin[\[Pi] E];
S[k_] := S[k] = S[k - 1] + Block[{$MaxExtraPrecision = 10000}, N[Sin[\[Pi] E k!], 20]];
pd = Table[{k, S[k]}, {k, 0, 2000}];
ListPlot[pd, PlotRange -> {0, 2}]

The series appears to converge to S[2000]=1.021106473303102347