To determine the convergence or divergence of the series.

\(\sum_{n=1}^\infty\frac{n}{(n+1)2^{n-1}}\)

let

\(a_n=\frac{n}{(n+1)2^{n-1}}\)

\(b_n=\frac{1}{2^n}\)

By limit comparison test

let \(\sum a_n\) and \(\sum b_n\) be two series such that

\(\lim_{n\to\infty}\frac{a_n}{b_n}=c\quad0\)

then both series will converge or diverge together.

\(\lim_{n\to\infty}\frac{a_n}{b_n}=\lim_{n\to\infty}\frac{\frac{n}{(n+1)2^{n-1}}}{\frac{1}{2^n}}\)

\(=\lim_{n\to\infty}\frac{2n}{n+1}\)

\(=2\)

which is non zero and finite

Hence, both series converge or diverge together.

Since, \(\sum b_n\) is a geometric series with common ratio les than 1

Thus, it converges

Hence, by comparison test \(\sum a_n\) converges