Evaluate $\sum_{n=1}^{\infty} \frac{\ln n}{(n+1)!}$

Question

Is there a closed form expression for this sum (or very similar sums) with known transcendental functions or irrational numbers? $$\sum_{n=1}^{\infty} \frac{\ln n}{(n+1)!} \approx 0.17549799405696\dots$$ Related approximate expressions would also be appreciated.

Answer 1

Well, if you introduce a parameter, like so:

$$\sum_{n=0}^\infty \frac{\log(n+x)}{(n+1)!},$$ and differentiate it with respect to $x,$ you will get a sum in terms of incomplete Gammas, which you can then integrate back. The sad part, of course, that you still have to be able to sum this for SOME $x$ to get the constant of integration.