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$$\sum _{n=0}^{\infty \:}\left(n\ e^{-n^2}\right)$$

Can I still use the integral test to determine whether this series converges or diverges given that $f(x) = x\ e^{-x^2}$ is not decreasing on the interval $(0,\frac{\sqrt{2}}{2}]$?

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Yes, the important thing is that it is decreasing after a while at least.

This is because a finite number of terms isn't relevant for the convergence of the serie.

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