real analysis - Suppose $sum{a_n}$ converges. Either prove that $sum{b_n}$ converges or give a counter-example

Suppose $\sum{a_n}$ is a convergent series of real numbers. Either prove that $\sum{b_n}$ converges or give a counter-example, when we define $b_n$ by:

1. $a_n \sin(n)$


2. $n^{\frac{1}{n}}a_n$

For the first one, I was thinking of using the fact that $|\sin(n)| \leq 1$ and then using comparison test. However, we don't know that $\sum{|a_n|}$ converges.

For the second one, I was thinking of using the fact that $\lim_{n\rightarrow \infty} n^{\frac{1}{n}}=1$. But, I'm completely stuck.

Thanks!

Answer

For 1), take $a_n = \dfrac{\sin n}{n}$. Then $\sum a_n$ converges (by Dirichlet's test), but

$$\sum \frac{\sin^2 n}{n}$$
diverges, see Convergence of $\sum_{n=1}^\infty \frac{\sin^2(n)}{n}$.

For 2), the series $\sum b_n$ converges by Abel's test