Convergent series of positive terms

Question

I got stuck on this problem while preparing for an exam:

Let $\sum_{n=1}^\infty a_n$ be a convergent series of positive terms. Show that $\sum_{n=1}^\infty a_n^{\frac{n}{n+1}}<\infty$.

There is a hint that suggests examining the set $S=\{n : a_n^{1/(n+1)}\leq 1/2\}$ but I am not sure how this helps. I was thinking of partitioning $\mathbb{N}$ into countably many sets by changing the fraction in $S$ but I am not sure if this is the right direction to go. I would just love a hint to get started. Thanks in advance

just to make sure, in your definition of $S$ the exponent is $1/n + 1$ or $1/(n+1)$? — D. Ungaretti, Aug 10 '19 at 23:33

score 9 · Accepted Answer · answered Aug 10 '19 at 23:37

9

If $a_n^{\frac 1 {n+1}} \leq \frac 1 2$ then $a_n^{\frac n {n+1}}\leq \frac 1 {2^{n}}$ and if $a_n^{\frac 1 {n+1}} >\frac 1 2$ then $a_n^{\frac n {n+1}} \leq 2 a_n$. Hence $a_n^{\frac n {n+1}} \leq 2a_n+\frac 1 {2^{n}}$ for all $n$. From this it follows that $\sum a_n^{\frac n {n+1}} <\infty$.

answered Aug 10 '19 at 23:37

Kavi Rama Murthy

311,013

Amazing proof! +1 – xbh Aug 10 '19 at 23:42

Convergent series of positive terms

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