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 Sep 5 comment Upper bound for an infinite series with a square root That's great, thanks. I'd come across an integral version before (Gradshteyn and Ryzhik, 12.411), but hadn't realised it was generalisable in this way. I come from an engineering background, so my knowledge of the connections between these things is sometimes a bit limited. Sep 5 comment Upper bound for an infinite series with a square root That's an excellent result. I had suspected the solution (on a hunch), but I couldn't think of how to show it. Also, I've never come across the series form of Jensen's inequality before, so thanks for that too. Sep 5 awarded Supporter Sep 5 accepted Upper bound for an infinite series with a square root Sep 4 revised Upper bound for an infinite series with a square root Fixed the inequality so that it works for $k \geq 0$ rather than just $k > 0$ (the range wasn't specified previously). Sep 4 asked Upper bound for an infinite series with a square root Dec 21 revised Sum of an infinite series with regularized gamma functions Edited to make the dependency of the inner series on the index of the outer series explicit. Dec 21 comment Sum of an infinite series with regularized gamma functions Thanks, Johnny! I'll persevere... Dec 20 awarded Commentator Dec 20 comment Sum of an infinite series with regularized gamma functions For example, if you expand my original expression, you get: $$S = e^{-x-y} \left(x + \frac{x^2}{2} (1 + y) + \frac{x^3}{6}(1 + y + \frac{y^2}{2}) + \cdots + \frac{x^n}{n!}(1 + y + \cdots + \frac{y^{n - 1}}{(n-1)!}) + \cdots \right)$$ Dec 20 awarded Cleanup Dec 20 comment Sum of an infinite series with regularized gamma functions EDIT: I've reverted your edit of my question to the original. Dec 20 revised Sum of an infinite series with regularized gamma functions rolled back to a previous revision Dec 20 comment Sum of an infinite series with regularized gamma functions Thanks again, but I think you might misunderstand what I wrote. The expression I want to evaluate is exactly as I wrote earlier. You can't simply replace the outer series with the lower regularized Gamma function, and the inner one with the upper regularized Gamma function, because the upper limit of the inner series has a dependency on the index of the outer one. It's not simply the product of two separable functions. Dec 20 comment Sum of an infinite series with regularized gamma functions Yes, the outer series is indexed by $k$, which ranges from $1$ to $\infty$, but the inner series is indexed by $l$, which ranges from $0$ to $k - 1$. Dec 20 comment Sum of an infinite series with regularized gamma functions Thanks, but I think you missed the dependency of the second series on $k$, i.e. $\sum_{l=0}^{k-1} \cdots$. Dec 20 accepted Advice on an integral involving the error function Dec 20 revised Sum of an infinite series with regularized gamma functions The outer series should start at k = 1, not k = 0. Dec 20 asked Sum of an infinite series with regularized gamma functions Dec 10 awarded Teacher