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Question # Use the Limit Comparison Test to determine the convergence or divergence of the series. sum_{n=1}^inftyfrac{n}{(n+1)2^{n-1}}

Series
ANSWERED Use the Limit Comparison Test to determine the convergence or divergence of the series.
$$\sum_{n=1}^\infty\frac{n}{(n+1)2^{n-1}}$$ 2021-01-29

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