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It is known that the power series $\sum_{n=0}^\infty a_n x^n$ and $\sum_{n=0}^\infty n a_n x^{n-1}$ have the same radius of convergence $r$. Is it true that if $r<\infty$ and $\sum_{n=0}^\infty na_n r^{n-1}$ converges then $\sum_{n=0}^\infty a_n r^n$ converges, too? The same question for $-r$. If not, what is a counterexample.

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    $\begingroup$ With an obvious change of variable, this amounts to the question without power series: if $\sum b_n$ converges, does $\sum b_n/n$ converge? $\endgroup$
    – YCor
    Dec 14, 2017 at 0:45
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    $\begingroup$ And it seems to be true, by Abel's summation formula. (Beware that we can have $\sum |b_n/n|=\infty$, e.g., when $b_n=(-1)^n/\log(n)$ where $\sum b_n$ converges.) $\endgroup$
    – YCor
    Dec 14, 2017 at 0:57
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    $\begingroup$ ... or use Dirichlet's test. You can replace $1/n$ by any sequence decreasing monotonically to $0$, and you don't even need $\sum b_n$ to converge, just that the partial sums are bounded. $\endgroup$ Dec 14, 2017 at 2:07
  • $\begingroup$ Thanks! Now I understand why this fact is true. By the way, can one see this fact with a proof somewhere in the literature? $\endgroup$
    – owb
    Dec 14, 2017 at 15:47
  • $\begingroup$ Abel summation? it sounds hard not to find a proof... $\endgroup$
    – YCor
    Dec 15, 2017 at 3:15

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