# Convergence of $\sum_{n=1}^{\infty}\frac{1}{n}\sum_{k=1}^{\infty}e^{-(n-k)^{2}}$

This series $\sum_{n=1}^{\infty}\frac{1}{n}\sum_{k=1}^{\infty}e^{-(n-k)^{2}}$ converges? I think not, I used the integral: $\int_{1}^{\infty}\int_{1}^{\infty}\frac{1}{x}e^{-(x-y)^{2}}dxdy\rightarrow \infty$ then the series diverges. What do you think?

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A lot simpler than that! Look at inner sum, put $k=n$. You get $1$. So inner sum is bigger than $1$. Then it is comparison with harmonic series. – André Nicolas Mar 2 '12 at 23:42

## 1 Answer

HINT: $$\sum_{n=1}^m \frac{1}{n}\sum_{k=1}^\infty \exp(-(n-k)^2) > \sum_{n=1}^m \frac{1}{n} \sum_{k=1}^\infty \delta_{k,n} \exp(-(n-k)^2) = \sum_{n=1}^m \frac{1}{n}$$

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Upvoted! Would you mind giving this a try? – Pedro Tamaroff Mar 2 '12 at 22:42