how to evaluate the following double summation to infinity without using integration method?

The expression is as follows:

$\sum_{x=0}^{\infty}\sum_{y=0}^{\infty} \exp(-\sqrt{x^2+y^2})$

I have thought about using Taylor approximation to get started but it doesn't seem to get me anywhere.

Any hints on how to get started will be appreciated. Thank you in advance.

I assume that $x$ and $y$ are integral, in which case your sum is one quarter of the sum
$$\sum_{r=1}^\infty \exp(-\sqrt{r}) s(r),$$ where $s(r)$ is the number of representations of $r$ as the sum of two squares. Clearly, $\sum_{r=1}^R s(r) \sim \pi r,$ which means you can estimate the original sum via summation by parts, followed by approximating the sum by the integral.
• In your notation the area of a circle is $\pi r$. The sum you get with the aporoximation can be computed exactly by summation by parts again. – Will Sawin May 16 '15 at 23:56