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If $x_j$ is a positive sequence, what is the condition on $x_j$ so that $$\sum_{j=1}^\infty \exp(-x_j^2)$$ converges?

EDIT: A general condition doesn't exist. My question therefore becomes: if $x_j^2\geq j^{1/r}$ for some positive $r$, does the above series converge?

sjage
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  • Any thoughts on the problem? What do you know about sufficient conditions for convergence? – saz Oct 05 '17 at 19:56
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    I know about the ratio test and if I'm not wrong this gives the condition $x_{j+1}^2>x^2_j$ – sjage Oct 05 '17 at 20:07
  • Frankly I am not sure what you expect from this. Any sequence $(a_j)$ with $0< a_j < 1$ can be written in the form $a_j = e^{-x_j^2}$ for some positive $x_j$, so you are essentially asking when such $\sum_j a_j$ converges or not. Of course, there cannot be such a criteria unless you focus on very specific situations. – Sangchul Lee Oct 05 '17 at 20:08
  • The condition does not exist. There is no universal convergence criterion. –  Oct 05 '17 at 20:19

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From a certain $j$ on, $e^{-x_{j}^2}$ should be less or equal to the "laziest" converging function, which among those at constant power is $j^{\,-(1+\delta)}$.

Thus $\quad e^{-x_{j}^2} \le j^{\,-(1+\delta)} \quad \Rightarrow \quad x_{j}^2 > \ln(j)$
So if you want a limit with a power of $j$, you have $ \ln(j)<j<x_{j}^2$.

G Cab
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