Does the following series have a closed form ?
$$\sum_{n=1}^{+\infty}\frac{\mathrm{e}^{-\sqrt{n}}}{\sqrt{n}}$$
Motivation :
The original exercise is
Compute $\int_{1}^{+\infty} \sum_{n=1}^{+\infty} \exp(-x\cdot \sqrt{n})dx$
I switched sums and integral using my course. But I am stuck to compute the series.
If it does, then it's not known. When the exponent itself is a power of the variable, the only known closed forms are for when its exponent is $1$ or $2$; e.g., $\displaystyle\sum_{n=-\infty}^\infty e^{-\pi n^2}=\frac{\sqrt[4]\pi}{\Gamma\bigg(\dfrac34\bigg)}$ , to give a less trivial example. However, the definite integral associated with our series does have a rather nice closed form, namely $~\displaystyle\int\frac{e^{-\sqrt x}}{\sqrt x}dx=2\int e^{-t}~dt=-2e^{-t}=-2e^{-\sqrt x}$. When computed over $(1,\infty)$, it yields $\dfrac2e$ as a result. Indeed, the numerical value of our series is slightly bigger than that, around $1$.