use the root test to determine if the following series converges absolutely or diverges\n sum _ { n = 1 } ^…

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}). 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\rightarrow\infty}\sqrt[n]{7}=1) (because for any positive real - number (a), (\lim_{n\rightarrow\infty}\sqrt[n]{a}=1)), then (\lim_{n\rightarrow\infty}\sqrt[n]{\vert a_{n}\vert}=\lim_{n\rightarrow\infty}\frac{\sqrt[n]{7}}{4n + 5}).

Step3: Evaluate the limit

We know that (\lim_{n\rightarrow\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\rightarrow\infty}\frac{f(n)}{g(n)} = 0), here (f(n)=1) and (g(n)=4n + 5)). So (\lim_{n\rightarrow\infty}\sqrt[n]{\vert a_{n}\vert}=0).

Since (\lim_{n\rightarrow\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.