Chapter 13.4, Problem 2E

To determine

To fill: The definition for the limit of the sequence $a_1,a_2,a_3,\cdots$.

Answer to Problem 2E

When the nth term $a_n$ of the sequence $a_1,a_2,a_3,\cdots a_n$ with limit L has the value close to L where n is sufficiently large. Then, the limit of sequence converges; otherwise, the sequence diverges.

Explanation of Solution

Definition1:

For a sequence $a_1,a_2,a_3,\cdots$ with the limit L

If, the nth term $a_n$ of the sequence $a_1,a_2,a_3,\cdots a_n$ is taken arbitrarily close to L where n to be sufficiently large. The expression becomes, $\underset{n\to \infty }{\lim }{a}_n$.

Mathematically it can be explained as,

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

Where, $a_n\to L\text{as}n\to \infty$

Thus when the limit exists, the sequence converges; otherwise, the sequence diverges.