Use the Ratio Test to determine the convergence or divergence of the series. $\sum _ { n = 1 } ^ { \infty } n \left( \frac { 8 } { 5 } \right) ^ { n }$
A)Ratio Test is inconclusive
B)diverges
C)converges

Question

Quizplus · Accepted Answer

The Ratio Test involves taking the limit of the absolute value of the ratio of the (n+1)th term to the nth term as n approaches infinity. For the given series, the (n+1)th term is (n+1)(8/5)^(n+1) and the nth term is n(8/5)^n. The ratio is |[(n+1)(8/5)^(n+1)] / [n(8/5)^n]| = |(n+1)/n * (8/5)|. As n approaches infinity, this ratio approaches 8/5, which is greater than 1. According to the Ratio Test, if the limit of this ratio is greater than 1, the series diverges.

Use the Ratio Test to Determine the Convergence or Divergence ∑n=1∞n(85)n\sum _ { n = 1 } ^ { \infty } n \left( \frac { 8 } { 5 } \right) ^ { n }∑n=1∞​n(58​)n

Use the Ratio Test to Determine the Convergence or Divergence $\sum _ { n = 1 } ^ { \infty } n \left( \frac { 8 } { 5 } \right) ^ { n }$