Chapter 8, Problem 18RQ

To determine

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

Answer to Problem 18RQ

Thestatement is true as the ratio test is passed.

Explanation of Solution

Given:

Thegiven statement is that if $a_n>0$ and $\underset{n\to \infty }{\text{lim}}\left(\frac{a_{n+1}}{a_n}\right)<1$ , then $\underset{n\to \infty }{\text{lim}}{a}_n=0$ .

Calculation:

Consider ${\displaystyle \sum a_n}$ converges, then by the ratio test.

  $\underset{n\to \infty }{\lim }\left|\frac{a_n+1}{a_n}\right|<1$

If $\left\{a\right\}$ be a sequence such that,

  $\underset{n\to \infty }{\lim }\left|\frac{a_n+1}{a_n}\right|=l$

Here, $\left|l\right|<1$ .

Then,

  $\underset{n\to \infty }{\lim }{a}_n=0$

Thus, the statement is true as the ratio test is passed.