Chapter 8.1, Problem 2E

(a)

To determine

To Find: The definition of the convergent sequence and two examples for it.

Answer to Problem 2E

The definition of the convergent sequence and two examples for it are obtained.

Explanation of Solution

Calculation:

Consider the series with the $n$ term is convergent only when $\underset{x\to \infty }{\lim }{a}_n$ exists.

Example 1

  $a_n=\frac{n}{n+1}$

In the above expression as $a_n\to 1$ then, $n\to \infty$ .

Example 2

  $a_n=\frac{1}{2^n}$

In the above expression as $a_n\to 0$ then, $n\to \infty$ .

(b)

To determine

To Find: The definition of the divergent sequence and two examples for it.

Explanation of Solution

Calculation:

Consider the series with the $n$ term is divergent only when $\underset{x\to \infty }{\lim }{a}_n$ do not exists.

Example 1

  $a_n=n$

In the above expression as $a_n\to \infty$ then, $n\to \infty$ .

Example 2

  $a_n=\frac{n^3}{n^2-5}$

In the above expression as $a_n\to \infty$ then, $n\to \infty$ .