Answer
$\Sigma_{n=1}^{\infty} (\frac{n+1}{n})a_{n}$ is absolutely convergent.
Work Step by Step
It is given that $\Sigma_{n=1}^{\infty} a_{n}$ is absolutely convergent , which means that the series $\Sigma_{n=1}^{\infty} |a_{n}|$ converges.
Consider, $\Sigma_{n=1}^{\infty} |(\frac{n+1}{n})a_{n}|$
We have $\lim\limits_{n \to \infty}\frac{ |(\frac{n+1}{n})a_{n}|}{ |a_{n}|}=\lim\limits_{n \to \infty}|\frac{n+1}{n}|=1$
It follows that the given series behaves like $\Sigma_{n=1}^{\infty} |a_{n}|$ and thus converges. So, we conclude that the series $\Sigma_{n=1}^{\infty} (\frac{n+1}{n})a_{n}$ is absolutely convergent.
