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I wish to find an explicit expression for the following function serie: $$ \sum_{i=0}^{\infty} 2^{id} \exp(-|x|^2 2^i), $$ where $d$ is a small integer ($d=1,2,3$). It is clear that the series converges pointwise in $ \mathbb{R}\setminus\{0\}$ and it converges uniformly in $ \mathbb{R}\setminus B_r(0)$ for any positive $r$, from the Weierstrass M-test and the ratio test.

I would like to have the explicit result of the serie (as a function of $x$).

Thank you for your help.

edop
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    You must have meant uniform convergence out of a neighborhood of $x=0$ ('cause the series diverges there). As the sum is $(-1)^d f^{(d)}(x^2)$ where $f(x)=\sum_{n=0}^{\infty}\exp(-2^n x)$, it goes this way. Nothing to expect (for a closed form). – metamorphy Oct 01 '19 at 08:35
  • You are right about the convergence domain. I edited the question. – edop Oct 02 '19 at 09:44

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