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I was doing various tasks about convergence/divergence of series, where i had to use various theorems, but here i don't have any numbers, just general series.

So i have problem with two of them.

We know that $a_n$ is positive and $\sum a_n$ is convergent. Is that true $\sum a_n\sqrt[4]{a_n}$ converges too? Is it true that $\sum \frac{a_n}{\sqrt{n}}$ converges too?

In first task i was trying to show that it diverges, but i couldn't find any series to get into comparison test. In second one i have no clue how to show it, because, to my intuition, if $a_n$ converges and $a_n$ has only positive numbers, then smaller number converges too.

Any help? Thanks in advance!!!

Krzysztof Lewko
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1 Answers1

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Hints: $\lim_{n\to\infty}a_n=0$, hence $0\leqslant a_n\leqslant 1$ for $n$ large enough and $a_n\sqrt[4]{a_n}\leqslant a_n$.

We have $n^{-1/2}\leqslant 1$ for each $n$ hence we conclude similarly.

Davide Giraudo
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