Question

Use the Root Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{n}{4^{n}} $$

Short answer

The series \( \sum_{n=1}^{\infty} \frac{n}{4^{n}} \) is convergent according to the Root Test.

Step-by-step solution

Identifying the terms of the series

In the given series \( \sum_{n=1}^{\infty} \frac{n}{4^{n}} \), the nth term is \(a_n = \frac{n}{4^n}\).

Applying the Root Test

In the Root Test, calculate the limit as n approaches infinity of the nth root of the absolute value of the nth term, i.e., \( \lim_{n \to \infty} \sqrt[n]{|a_n|} \). Here, \( a_n = \frac{n}{4^n} \) so \( \sqrt[n]{|a_n|} = \sqrt[n]{\frac{n}{4^n}} = \left( \frac{n}{4^n} \right)^{1/n} = \frac{n^{1/n}}{4} = \frac{1}{4}\cdot n^{1/n} \).

Calculating the limit

Next, calculate \( \lim_{n \to \infty} \frac{1}{4}\cdot n^{1/n} \). As n approaches infinity, \( n^{1/n} \) approaches 1, so the limit is \( \frac{1}{4} \).

Determining the convergence

Since the limit \( L=\frac{1}{4} < 1 \), according to the Root Test, the series \( \sum_{n=1}^{\infty} \frac{n}{4^{n}} \) is convergent.