Question

Use the Ratio Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} n\left(\frac{3}{2}\right)^{n} $$

Short answer

The series is divergent.

Step-by-step solution

Express the series

Firstly, express the series as \(a_n = n\left(\frac{3}{2}\right)^{n}\). The Ratio Test involves taking the ratio of \(a_{n+1}\) and \(a_n\).

Find the ratio \(a_{n+1}/a_n\)

Calculate the ratio \(a_{n+1}/a_n\), where \(a_{n+1} = (n+1)\left(\frac{3}{2}\right)^{n+1}\) and \(a_n = n\left(\frac{3}{2}\right)^{n}\). So the ratio becomes \(\frac{(n+1)\left(\frac{3}{2}\right)^{n+1}}{n\left(\frac{3}{2}\right)^{n}} = \frac{3(n+1)}{2n}\).

Find the limit

Next, we find the limit of the ratio as \(n\) approaches infinity using the rule \( \lim_{n \rightarrow \infty } \frac{3(n+1)}{2n}\). This simplifies to 1.5, since the coefficients of \(n\) in the numerator and the denominator are the same.

Interpret the result

Since 1.5 is greater than 1, per the Ratio Test, it can be concluded that the series is divergent.