I am interested in finding the approximate answer to the following infinite sum


\begin{equation}
\sum\_{l=0}^{\infty}( l+a) \exp^{b{(l+c)}^2}
\end{equation}
in the case where $a>0 , b<0 , c>0$ and $a$ and $c$ are both large. Does any one know how to approach this problem and where to look for methods if I need to find the asymptotic answer to infinite sums?



Approximating an infinite Sum
Approximating the sum by an integral, gives
$$F(a,b,c)=
\int\_0^\infty( l+a) \exp\left[b{(l+c)}^2\right]\,dl=\frac{\sqrt{\pi } (a-c) \text{Erfc}\left(c\sqrt{-b} \right)}{2\sqrt{-b}}-\frac{e^{b c^2}}{2b}.$$
This should be accurate for large $c$. Here is a comparison for $b=-1$, $c=2a$ of the sum (blue) and the integral approximation (gold) as a function of $a$.


![](https://ilorentz.org/beenakker/MO/Fabc.png)


