Chapter 10.4, Problem 10QR

To determine

To identify the series and find the smallest positive integer $N$ .

Answer to Problem 10QR

  $a_n=\frac{1}{n^2},b_n=\frac{1}{n^3}$ and $N=2$

Explanation of Solution

Given:

The given series are ${\displaystyle \sum \frac{1}{n^2}},{\displaystyle \sum n^{-3}}$

Calculation:

The given series are,

  ${\displaystyle \sum \frac{1}{n^2}},{\displaystyle \sum \frac{1}{n^3}}$

Using a calculator to graph the function, we get $\frac{1}{n^2}>\frac{1}{n^3}$ for $n>1$ .

Taking $a_n=\frac{1}{n^2},b_n=\frac{1}{n^3}$ and $N=2$ . We get $a_n>b_n$ for all $n\ge N$ .

Therefore,

  $a_n=\frac{1}{n^2},b_n=\frac{1}{n^3}$ and $N=2$