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I feel that this general fact should be known. Suppose I have a strictly increasing sequence of integer positive numbers $\{a_n\}_{n \in \mathbb{N}}$. We want to investigate the behavior of $$ f(x)= \sum_{n=0}^{\infty} x^{a_n}.$$ Of course this is a power series with convergence radius $R=1$. I want to know something about the limit $\lim_{x \to 1^-} (1-x)^{\alpha} f(x)$, depending on $\alpha$. Are there Tauberian/Abelian theorems which can apply? This comes from a particular problem when $a_n=n^2$ where I think I can show that $$\lim_{x \to 1^-} \sqrt{1-x} \cdot f(x) =\int_0^{\infty} e^{-x^2} \,dx = \frac {\sqrt{\pi}}2,$$ by looking at it as an approximated Riemann sum. I am interested also at behaviors of other similar series like $f(x)= \sum_{p \; prime} \frac{x^p}p$ etc...

StheW
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  • Related : https://math.stackexchange.com/questions/1196721/proving-that-lim-x-to1-left-sqrta1-x-cdot-sum-n-0-inftyxna-ri – jvc Feb 22 '21 at 18:51
  • Thank you very much! I looked for similar posts but I couldn't find them. I would say that this extends easily to the case of polynomials. I found also the case $a_n=2^n$ which is interesting. How about $a_n=n!$ or $a_n=p$ with p prime? – StheW Feb 23 '21 at 08:00
  • I am not really familiar with theses sums... I hope the others will be. I am also interested by the case $2^n$. Where did you find it? – jvc Feb 23 '21 at 09:34
  • There are several answers already, here's one: https://math.stackexchange.com/questions/296066/evaluate-lim-x-to1-left-sum-n-0-infty-leftx2n-right-log-2-f?rq=1 – StheW Feb 28 '21 at 06:37
  • Using a very similar approach to the $n^2$ case, one should look at the convergence of the measures $\mu_k = \frac 1{A_k} \sum_{i=0}^{\infty} \delta_{a_n/k}$, where $A_k= { n : a_n \leq k}$. Then if $\mu_k \rightharpoonup \mu$ for $k \to \infty$ morally one should have $\frac{f(1-1/k)}{A_k} \to \int_0^{\infty} e^{-x} , d \mu$. – StheW Mar 02 '21 at 12:11

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