TITLE: Closed form of the sum $\sum\limits_{n=0}^\infty \exp(-n^3)$
QUESTION [6 upvotes]: I am trying to calculate the sum of the series
$$\sum_{n=0}^\infty \exp(-n^3)$$
Can it be expressed in terms of known mathematical functions?

REPLY [5 votes]: I guess the answer ("it is hard") have been given. 
The numerical computation gives
$\sum_{n=0}^\infty{\mathrm e}^{-n^3}=1.368\dots$
I want to add that a natural first order approximation given by the Euler–Maclaurin formula, is
$$\sum_{n=0}^\infty{\mathrm e}^{-n^a}\approx\frac{1}{2}+\Gamma\left(1+\frac{1}{a}\right)\approx 1.5-\frac{\gamma}{a},$$
where $\gamma=0.577\dots$ is the Euler–Mascheroni constant. 
The approximation is best at about $a\approx 4.5$.
For you with $a=3$ we find 
$\frac{1}{2}+\Gamma\left(1+\frac{1}{3}\right)=1.392\dots$