proving/disproving the statements

Question

I would like to, if possible , to receive assistance, in layman terms, in the proving/disproving of the following statements:

$ 1)$ If the positive series $ \sum_{n=1}^{\infty}na_{n} $ converges, then $ \sum_{n=1}^{\infty}na_{n+1} $ converges as well.

$ 2)$ If $ \sum_{n=1}^{\infty}a_{n} $ converges, and $ \sum_{n=1}^{\infty}b_{n} $ converges absolutely, then $ \sum_{n=1}^{\infty}a_{n}b_{n} $ converges .

Answer 1

For 1):

You can define $s_n = \sum_{k=1}^{n} k a_k$.

It is clear that $(s_n)_{n\in\mathbb{N}}$ converges. Now:

$$ \sum_{k=1}^{n} k a_{k+1}=\sum_{k=1}^{n} \frac{k}{k+1}(s_{k+1} - s_k) = \sum_{k=1}^{n+1} \frac{k-1}{k} s_k - \sum_{k=1}^{n} \frac{k}{k+1} s_k =\\ \frac{n}{n+1} s_{n+1} - \sum_{k=1}^{n} \frac{1}{k(k+1)}s_k.$$

It follows that the last expression converges as $n\to\infty$.

For 2):

Convergent implies bounded.

If $a_n\le c$ then $\sum a_nb_n\le \sum cb_n\le c\sum b_n$, and the latter converges.