## Calculus 8th Edition

$\Sigma_{n=1}^{\infty} (\frac{n+1}{n})a_{n}$ is absolutely convergent.
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 behave like $\Sigma_{n=1}^{\infty} |a_{n}|$that is converges. So, we conclude that the series is $\Sigma_{n=1}^{\infty} (\frac{n+1}{n})a_{n}$ is absolutely convergent.