Use the Ratio Test to determine the convergence or divergence of the series. $$\sum _ { n = 1 } ^ { \infty } n \left( \frac { 3 } { 10 } \right) ^ { n }$$
A)diverges
B)Ratio Test inconclusive
C)converges

Question

Quizplus · Accepted Answer

The Ratio Test involves taking the limit as $n$ approaches infinity of the absolute value of the ratio of the $(n+1)$th term to the $n$th term of the series. For the given series, the $(n+1)$th term is $(n+1)\left(\frac{3}{10}\right)^{n+1}$ and the $n$th term is $n\left(\frac{3}{10}\right)^n$. The ratio of these terms is $\frac{(n+1)\left(\frac{3}{10}\right)^{n+1}}{n\left(\frac{3}{10}\right)^n} = \frac{3}{10} \cdot \frac{n+1}{n}$. As $n$ approaches infinity, this ratio approaches $\frac{3}{10}$, which is less than 1. According to the Ratio Test, if the limit of the absolute value of this ratio is less than 1, the series converges.

Use the Ratio Test to Determine the Convergence or Divergence ∑n=1∞n(310)n\sum _ { n = 1 } ^ { \infty } n \left( \frac { 3 } { 10 } \right) ^ { n }n=1∑∞​n(103​)n

Use the Ratio Test to Determine the Convergence or Divergence $\sum _ { n = 1 } ^ { \infty } n \left( \frac { 3 } { 10 } \right) ^ { n }$