Chapter 8, Problem 20RQ

To determine

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

Answer to Problem 20RQ

Therequired statement is true as $\underset{n\to \infty }{\text{lim}}{a}_{n+3}-\underset{n\to \infty }{\text{lim}}{a}_n$ is equal to 0.

Explanation of Solution

Given:

Thegiven statement is that if $\underset{n\to \infty }{\text{lim}}{a}_n=2$ then $\underset{n\to \infty }{\text{lim}}\left(a_{n+3}-a_n\right)=0$ .

Calculation:

Consider the given information is,

  $\underset{n\to \infty }{\text{lim}}{a}_n=2$

Then, if the nth term converges to 2. Then the terms after it will converge to 2 and the $n+3$ term will converge to 2.

Then, subtract the two equations and obtain the difference between the limits as,

  $\begin{array}{c}\underset{n\to \infty }{\text{lim}}{a}_{n+3}-\underset{n\to \infty }{\text{lim}}{a}_n=\underset{n\to \infty }{\text{lim}}\left(a_{n+3}-a_n\right)\\ =2-2\\ =0\end{array}$

Thus, the required statement is true as $\underset{n\to \infty }{\text{lim}}{a}_{n+3}-\underset{n\to \infty }{\text{lim}}{a}_n$ is equal to 0.