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Could anyone please give me some hints ?

Let $\sum a_n$ be a convergent serie of real number. Prove or disprove that $\sum a_n \sin n$ and $\sum n^{\frac{1}{n}} a_n$ are also convergent.

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  • $\begingroup$ 1. Try to play with sign of $\sin n$ and relatively convergence series. $\endgroup$ – Jihad Nov 17 '14 at 13:01
  • $\begingroup$ @mvggz This argument works if $a_n\ge0$. $\endgroup$ – Julián Aguirre Nov 17 '14 at 13:23
  • $\begingroup$ @ Julián Aguirre, indeed.. I have written that too quickly sorry $\endgroup$ – mvggz Nov 17 '14 at 13:27
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For the first one consider $\sum\sin n/n$. It converges by Dirichlet's test, but $\sum\sin^2n/n$ diverges (hint:$2\,\sin^2n=1-\cos(2\,n)$.)

For the second one use Abe's test ($n^{1/n}$ is monotone and bounded.)

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