Chapter 10.5, Problem 70E

To determine

To choose: The correct option for the given.

Answer to Problem 70E

The correct option is (E)

Explanation of Solution

Given information :

The series $4{\displaystyle \sum }_{n=1}^{\infty }\frac{1}{\sqrt{n}}$ , ${\displaystyle \sum }_{n=1}^{\infty }{\left(\frac{1}{\ln 4}\right)}^n$ , ${\displaystyle \sum }_{n=1}^{\infty }\frac{{(-1)}^{n+1}}{n^2}$

Calculation :

I. The series $4{\displaystyle \sum }_{n=1}^{\infty }\frac{1}{\sqrt{n}}$ diverges by the $p$ -Test.

II. The series ${\displaystyle \sum }_{n=1}^{\infty }{\left(\frac{1}{\ln 4}\right)}^n$ converges (geometric series with $r\approx 0.7)$..

III. The series ${\displaystyle \sum }_{n=1}^{\infty }\frac{{(-1)}^{n+1}}{n^2}$ converges absolutely by the $p$ -Test (or use the Alternating Series Test).

Option (E) is the correct answer