Properties of $\sum_{k=1}^{\infty} {\frac {{\ln}k} {k!}}$

Question

Consider the following infinite sum: $$ \sum_{k=1}^{\infty} {\frac {\ln k} {k!}} $$

(It is easy to show that it converges since ${\frac {\ln x} x}$ has a maximum.)

• Does it have a name?
• Can it be expressed in a closed form?
• Do we know if it's rational or not?

Answer 1

Let’s find an integral representation using @John Barber’s technique in this question. Now appears the digamma function:

\begin{align*}\sum_1^\infty \frac{\ln(n)}{n!} &=\int_1^\infty\lfloor x \rfloor \frac d{dx} \frac{\ln(x)}{x!}dx\\ &=\int_1^\infty \lfloor x\rfloor \left(\frac1{xx!}-\frac{\ln(x)ψ(x+1)}{x!}\right)dx\\ &= \int_1^\infty \frac{\lfloor x\rfloor}{xx!}dx -\int_1^\infty\frac{\lfloor x\rfloor\ln(x)ψ(x+1)}{x!}dx\\ &=-0.603782862791487988416183810982450548304170153164991021772413211382272284100525569478213750246…\end{align*}

Here is proof of the result.

Here is a visual representation of the constant. Here is an interactive graph too:

The “closed” form is nothing more than the $\text A306243$ constant unrelated to the above technique

equal to

$$\ln(\exp(1/2*\ln(2*\exp(1/3*\ln(3*\exp(1/4*\ln(4*\exp(...))))))))$$

Please correct me and give me feedback!