Question

use the root test to determine if the following series converges or diverges
sum_{n = 1}^{infty}\frac{-9}{(4+(1/n))^{3n}}
since the limit resulting from the root test is which is the root test
(simplify your answer type an exact answer)

Explanation:

Step1: Recall the root test formula

For a series \(\sum_{n = 1}^{\infty}a_{n}\), we calculate \(\lim_{n
ightarrow\infty}\sqrt[n]{\vert a_{n}\vert}\). Here, \(a_{n}=\frac{- 9}{(4+(1/n))^{3n}}\), so \(\vert a_{n}\vert=\frac{9}{(4+(1/n))^{3n}}\). Then \(\sqrt[n]{\vert a_{n}\vert}=\frac{\sqrt[n]{9}}{(4+(1/n))^{3}}\).

Step2: Calculate the limit

We know that \(\lim_{n
ightarrow\infty}\sqrt[n]{9}=1\) (since \(\lim_{n
ightarrow\infty}n^{k}=1\) for any constant \(k\)). And \(\lim_{n
ightarrow\infty}(4 +\frac{1}{n})=4\). So \(\lim_{n
ightarrow\infty}\sqrt[n]{\vert a_{n}\vert}=\frac{1}{4^{3}}=\frac{1}{64}\).

Answer:

Since the limit resulting from the root test is \(\frac{1}{64}\) which is \(< 1\), the root test indicates that the series converges.