use the root test to determine if the following series converges or diverges\n sum_{n = 1}^{infty}\frac{-9}{(…

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

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

Answer

Explanation:

Step1: Recall the root test formula

For a series (\sum_{n = 1}^{\infty}a_{n}), we calculate (\lim_{n\rightarrow\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\rightarrow\infty}\sqrt[n]{9}=1) (since (\lim_{n\rightarrow\infty}n^{k}=1) for any constant (k)). And (\lim_{n\rightarrow\infty}(4 +\frac{1}{n})=4). So (\lim_{n\rightarrow\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.