Is there a series satisfies $a_n>0,\sum a_n$ converges but $\sum a_n^{r}$ diverges for all $0\leq r<1$

Question

Is there a series satisfies $\displaystyle a_n>0 \forall n\in \mathbb{N},\sum_{n=1}^{\infty} a_n$ converges but $\displaystyle \sum_{n=0}^{\infty} a_n^{r}$ diverges for all $0\leq r<1$?

Context:

It originally comes out of a complex analysis problem where if I can find one I can find a series of holomorphic functions converges uniformly on a closed disc but not any open disc includes it.

There is no p-series satisfies this condition, and the series can not be something like geometric series either(otherwise by root test either both converge or both diverge).

Is such a series exists? If not I might approach my original problem wrong...

score 3 · Accepted Answer · answered Apr 10 '16 at 02:42

3

A start: Let $a_n=\frac{1}{(n+1)\ln^2 (n+1)}$.

answered Apr 10 '16 at 02:42

André Nicolas

514,336

Exactly what I was thinking. – marty cohen Apr 10 '16 at 03:02

Is there a series satisfies $a_n>0,\sum a_n$ converges but $\sum a_n^{r}$ diverges for all $0\leq r<1$

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