Chapter 9, Problem 41RE

To determine

To check: The series ${\displaystyle \sum _{n=1}^{\infty }-\frac{5}{n}}$ converges absolutely, converges conditionally, or diverges.

Answer to Problem 41RE

diverges

Explanation of Solution

Given information:

  ${\displaystyle \sum _{n=1}^{\infty }-\frac{5}{n}}$

Concept Used:

Absolutely Convergent: If the series ${\displaystyle \sum _{n=1}^{\infty }\left|a_n\right|}$ converges, then ${\displaystyle \sum _{n=1}^{\infty }{a}_n}$ converges absolutely.

Conditionally Convergent: If ${\displaystyle \sum _{n=1}^{\infty }{a}_n}$ converges, but the series ${\displaystyle \sum _{n=1}^{\infty }\left|a_n\right|}$ does not converges, then ${\displaystyle \sum _{n=1}^{\infty }{a}_n}$ converges conditionally.

Calculation:

Consider the series, ${\displaystyle \sum _{n=1}^{\infty }-\frac{5}{n}}$ .

Let, $a_n=\frac{5}{n}$

The series is diverged by using $p-$ test.

So, the series ${\displaystyle \sum _{n=1}^{\infty }-\frac{5}{n}}$ is diverges.