Question

use the root test to determine if the following series converges absolutely or diverges
\\(\sum_{n=1}^{\infty} \frac{6}{(4n + 7)^n}\\)

since the limit resulting from the root test is \\(\square\\), which is \\(\square\\) \\(\square\\)
(type an exact answer )

Explanation:

Step1: Recall Root Test Formula

The Root Test for a series \(\sum_{n = 1}^{\infty} a_n\) involves computing the limit \(L=\lim_{n
ightarrow\infty}\sqrt[n]{\vert a_n\vert}\). If \(L < 1\), the series converges absolutely; if \(L>1\), it diverges; if \(L = 1\), the test is inconclusive.

Step2: Identify \(a_n\)

For the series \(\sum_{n = 1}^{\infty}\frac{6}{(4n + 7)^n}\), we have \(a_n=\frac{6}{(4n + 7)^n}\). Then \(\vert a_n\vert=\frac{6}{(4n + 7)^n}\) (since it's positive for all \(n\geq1\)).

Step3: Compute \(\sqrt[n]{\vert a_n\vert}\)

$$ \sqrt[n]{\vert a_n\vert}=\sqrt[n]{\frac{6}{(4n + 7)^n}}=\frac{\sqrt[n]{6}}{\sqrt[n]{(4n + 7)^n}}=\frac{\sqrt[n]{6}}{4n + 7} $$

Step4: Compute the Limit as \(n

ightarrow\infty\)
We know that \(\lim_{n
ightarrow\infty}\sqrt[n]{6} = 1\) (since for any positive real number \(c\), \(\lim_{n
ightarrow\infty}\sqrt[n]{c}=1\)). And \(\lim_{n
ightarrow\infty}(4n + 7)=\infty\). So,

$$ \lim_{n ightarrow\infty}\frac{\sqrt[n]{6}}{4n + 7}=\frac{1}{\infty}=0 $$

Since \(L = 0<1\), the series converges absolutely.

Answer:

The limit resulting from the Root Test is \(0\), which is less than \(1\), so the series converges absolutely. The first box is \(0\), the first dropdown (comparison) is "less than", the second dropdown (conclusion) is "converges absolutely".