Problem 44 - Essential Calculus

Question

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

Step-by-Step Solution

Answer

The series is divergent.

Step 1: 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 \(a_n\).

Step 2: 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 \(\frac{(n+1)\left(\frac{3}{2}\right)^{n+1}}{n\left(\frac{3}{2}\right)^{n}} = \frac{3(n+1)}{2n}\).

Step 3: 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.

Step 4: Interpret the result

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