Chapter 8.4, Problem 38E

To determine

To calculate: Whether the given series is absolute convergent or not.

Answer to Problem 38E

The given series is not absolute convergent.

Explanation of Solution

Given information:

  ${\displaystyle \sum _{n=2}^{n=\infty }{\left(\frac{-2n}{n+1}\right)}^{5n}}$

Concept Used:

Root test: Whether the series is absolute convergent

  $\underset{n\to \infty }{\lim }\sqrt[n]{\left|a_n\right|}=L$

(i) If $L<1$ , then ${\displaystyle \sum a_n}$is absolute convergent.

(ii) If $L>1$ $(\text{or}L=\infty )$ , then ${\displaystyle \sum a_n}$is divergent.

(iii) If $L=1$, then the root test is inconclusive.

Calculation:

The series is ${\displaystyle \sum _{n=2}^{n=\infty }{\left(\frac{-2n}{n+1}\right)}^{5n}}$

By using the root test for this series to check it is absolute convergent.

  $\underset{n\to \infty }{\lim }\sqrt[n]{\left|a_n\right|}=L$

So, for the given series the $a_n$ is

  $\begin{array}{l}{a}_n={\left(\frac{-2n}{n+1}\right)}^{5n}\\ $ a_n={(-1)}^{5n}{\left(\frac{2n}{n+1}\right)}^{5n}$\end{array}$

This term $a_n$ is positive and negative for the value of $n\ge 1$ and its is alternating.

So,

  $\begin{array}{l}$ a_n={(-1)}^{5n}{\left(\frac{2n}{n+1}\right)}^{5n}$\left|a_n\right|=\left|{(-1)}^{5n}{\left(\frac{2n}{n+1}\right)}^{5n}\right|\\ ={\left(\frac{2n}{n+1}\right)}^{5n}\\ \left|a_n\right|={\left(\frac{2n}{n+1}\right)}^{5n}\end{array}$

Then put the value of $\left|a_n\right|$ in the root test equation

  $\begin{array}{l}L=\underset{n\to \infty }{\lim }\sqrt[n]{\left|a_n\right|}\\ =\underset{n\to \infty }{\lim }\sqrt[n]{{\left(\frac{2n}{n+1}\right)}^{5n}}\\ =\underset{n\to \infty }{\lim }{\left(\frac{2n}{n+1}\right)}^5\\ =\underset{n\to \infty }{\lim }\left(\frac{2}{1+\frac{1}{n}}\right)\\ =\left(\frac{2}{1+\frac{1}{n}}\right)\to 2asn\to \infty \\ \end{array}$

So,

  $\begin{array}{l}L=2asn\to \infty \\ L=2>1\end{array}$

According to root test if $L>1$ then ${\displaystyle \sum a_n}$ is divergent.

Means the given series is not absolute convergent.

Conclusion:

The given series is not absolute convergent.