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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.2012-03-02

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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](http://math.stackexchange.com/questions/106496/de-moivres-theorem-motivation-and-origins) a try?2012-03-02