Chapter 8, Problem 1RQ

To determine

To find: The statement is true or false and if the statement then given the explanation.

Answer to Problem 1RQ

The given stamen is false as ${\displaystyle \sum a_n}={\displaystyle \sum \frac{1}{n}}$ is divergent.

Explanation of Solution

Given:

The given statement is that $\underset{n\to \infty }{\lim }{a}_n=0$ , then ${\displaystyle \sum a_n}$ is convergent.

Calculation:

The given statement is that $\underset{n\to \infty }{\lim }{a}_n=0$ , then ${\displaystyle \sum a_n}$ is convergent is false as for ${\displaystyle \sum a_n}=\frac{1}{n}$ ,

  $\underset{n\to \infty }{\lim }\frac{1}{n}=0$

Then,

  ${\displaystyle \sum a_n}={\displaystyle \sum \frac{1}{n}}$

The above equation is divergent.