Chapter 8, Problem 4RCC

To determine

To find: The value of $\underset{n\to \infty }{\text{lim}}{a}_n$ and $\underset{n\to \infty }{\text{lim}}{s}_n$ .

Answer to Problem 4RCC

Thelimit converges to $\underset{n\to \infty }{\text{lim}}{a}_n=0$ .

Explanation of Solution

Given:

Thegiven value of ${\displaystyle \sum a_n}=3$ .

Calculation:

Consider the value of ${\displaystyle \sum a_n}=3$ this shows that ${\displaystyle \sum a_n}$ converges to 3.

Let $\langle S_n\rangle$ is the sequence of the partial sum of the series ${\displaystyle \sum a_n}$ .

Then,

  $S_n=a_1+a_2+\mathrm{....}{a}_{n-1}+a_n$

Also,

  $S_{n-1}=a_1+a_2+\mathrm{....}{a}_{n-1}$

Then, subtract the term as,

  $S_{n-1}-S_n=a_n$

Consider that as the series converges to 3 so $\langle S_n\rangle$ converges to 3.

Then,

  $\underset{n\to \infty }{\text{lim}}{S}_n=3$ And $\underset{n\to \infty }{\text{lim}}{S}_{n-1}=3$

Then, from $S_{n-1}-S_n=a_n$ and $\underset{n\to \infty }{\text{lim}}{S}_{n-1}=3$ ,

  $\begin{array}{c}\underset{n\to \infty }{\text{lim}}{a}_n=\underset{n\to \infty }{\text{lim}}\left(S_n-S_{n-1}\right)\\ =\underset{n\to \infty }{\text{lim}}{S}_n-\underset{n\to \infty }{\text{lim}}{S}_{n-1}\\ =3-3\\ =0\end{array}$

Thus, the limit converges to $\underset{n\to \infty }{\text{lim}}{a}_n=0$ .