Step by step solution to "Use the Limit Comparison Test to determine the..."

Tammy Todd

Answered question

2021-01-28

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}}

Answer & Explanation

Malena

Skilled2021-01-29Added 83 answers

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
$lim_{n \to \infty} \frac{a_{n}}{b_{n}} = c 0$
then both series will converge or diverge together.
$lim_{n \to \infty} \frac{a_{n}}{b_{n}} = lim_{n \to \infty} \frac{\frac{n}{(n + 1) 2^{n - 1}}}{\frac{1}{2^{n}}}$
$= lim_{n \to \infty} \frac{2 n}{n + 1}$
$= 2$
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

Jeffrey Jordon

Expert2021-12-27Added 2605 answers

Answer is given below (on video)

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