How do you use the ratio test to test the convergence of the series ∑ (-5)^(n+1)n / 2^n from n=1 to infinity?

Oct 17, 2015

See the explanation.

Explanation:

$L = {\lim}_{n \to \infty} | {a}_{n + 1} / {a}_{n} |$

$L = {\lim}_{n \to \infty} | \frac{{\left(- 5\right)}^{n + 2} \frac{n + 1}{2} ^ \left(n + 1\right)}{{\left(- 5\right)}^{n + 1} \frac{n}{2} ^ n} |$

$L = {\lim}_{n \to \infty} | \left(- 5\right) \frac{n + 1}{2 n} | = 5 {\lim}_{n \to \infty} \frac{n + 1}{2 n} = \frac{5}{2}$

$L > 1$ so the series is divergent.