Question

use the root test to determine if the following series converges absolutely or diverges.
sum _ { n = 1 } ^ { infty } \frac { 5 } { ( 8 n + 7 ) ^ { n } }
since the limit resulting from the root test is (square), which is
(type an exact answer.)

Explanation:

Step1: Recall the Root Test formula

For a series \(\sum_{n = 1}^{\infty}a_{n}\), we calculate \(L=\lim_{n
ightarrow\infty}\sqrt[n]{\vert a_{n}\vert}\). Here, \(a_{n}=\frac{5}{(8n + 7)^{n}}\), so \(\vert a_{n}\vert=\frac{5}{(8n + 7)^{n}}\).

Step2: Calculate \(\sqrt[n]{\vert a_{n}\vert}\)

\(\sqrt[n]{\vert a_{n}\vert}=\sqrt[n]{\frac{5}{(8n + 7)^{n}}}\). Using the property \(\sqrt[n]{\frac{m}{k^{n}}}=\frac{\sqrt[n]{m}}{k}\) (where \(m = 5\) and \(k=8n + 7\)), we get \(\sqrt[n]{\vert a_{n}\vert}=\frac{\sqrt[n]{5}}{8n + 7}\).
Since \(\lim_{n
ightarrow\infty}\sqrt[n]{5}=1\) (because for any positive real - number \(a\), \(\lim_{n
ightarrow\infty}\sqrt[n]{a}=1\)), then \(\lim_{n
ightarrow\infty}\sqrt[n]{\vert a_{n}\vert}=\lim_{n
ightarrow\infty}\frac{\sqrt[n]{5}}{8n + 7}\).

Step3: Evaluate the limit

We know that \(\lim_{n
ightarrow\infty}\frac{1}{8n + 7}=0\) (using the formula \(\lim_{n
ightarrow\infty}\frac{c}{an + b}=0\) for \(a>0\) and \(c,b\) constants). So, \(L = 0\).

Answer:

Since the limit resulting from the Root Test is \(0\), which is less than \(1\), the series \(\sum_{n = 1}^{\infty}\frac{5}{(8n + 7)^{n}}\) converges absolutely.