Question

use the root test to determine if the following series converges absolutely or diverges
sum _ { n = 1 } ^ { infty } \frac { 7 } { ( 4 n + 5 ) ^ { n } }
since the limit resulting from the root test is 0, which is greater than 1 the root test is inconclusive
(type an exact answer)

Explanation:

Step1: Recall the Root Test formula

For a series \(\sum_{n = 1}^{\infty}a_{n}\), we calculate \(\lim_{n
ightarrow\infty}\sqrt[n]{\vert a_{n}\vert}\). Let \(a_{n}=\frac{7}{(4n + 5)^{n}}\). Then \(\sqrt[n]{\vert a_{n}\vert}=\sqrt[n]{\frac{7}{(4n + 5)^{n}}}\).

Step2: Simplify the expression

Using the property \(\sqrt[n]{\frac{m}{k^{n}}}=\frac{\sqrt[n]{m}}{k}\) (where \(m = 7\) and \(k=4n + 5\)), we have \(\sqrt[n]{\vert a_{n}\vert}=\frac{\sqrt[n]{7}}{4n + 5}\).
Since \(\lim_{n
ightarrow\infty}\sqrt[n]{7}=1\) (because for any positive real - number \(a\), \(\lim_{n
ightarrow\infty}\sqrt[n]{a}=1\)), then \(\lim_{n
ightarrow\infty}\sqrt[n]{\vert a_{n}\vert}=\lim_{n
ightarrow\infty}\frac{\sqrt[n]{7}}{4n + 5}\).

Step3: Evaluate the limit

We know that \(\lim_{n
ightarrow\infty}\frac{1}{4n + 5}=0\) (using the fact that for a rational function \(\frac{f(n)}{g(n)}\) where \(\text{deg}(f(n))<\text{deg}(g(n))\), \(\lim_{n
ightarrow\infty}\frac{f(n)}{g(n)} = 0\), here \(f(n)=1\) and \(g(n)=4n + 5\)). So \(\lim_{n
ightarrow\infty}\sqrt[n]{\vert a_{n}\vert}=0\).

Since \(\lim_{n
ightarrow\infty}\sqrt[n]{\vert a_{n}\vert}=0<1\), by the Root Test, the series \(\sum_{n = 1}^{\infty}\frac{7}{(4n + 5)^{n}}\) converges absolutely.

The incorrect part in the original statement is that when the limit from the Root Test \(L = 0\), and according to the Root Test:

• If \(L<1\), the series \(\sum_{n = 1}^{\infty}a_{n}\) converges absolutely.
• If \(L>1\), the series \(\sum_{n = 1}^{\infty}a_{n}\) diverges.
• If \(L = 1\), the Root Test is inconclusive.

Answer:

Since the limit resulting from the Root Test is \(0\), which is less than \(1\), the series \(\sum_{n = 1}^{\infty}\frac{7}{(4n + 5)^{n}}\) converges absolutely.