Chapter D, Problem 24E

To determine

To prove:

  $\underset{n\to \infty }{\lim }{n}^3=\infty$ by using definition $4$ .

Answer to Problem 24E

it is proved that $\underset{n\to \infty }{\lim }{n}^3=\infty$ so.

Explanation of Solution

Given information:

  $\underset{n\to \infty }{\lim }{n}^3=\infty$

Proof:

Let’s assume notation as $a=n^3$

There is need to prove that $\underset{n\to \infty }{\lim }{n}^3=\infty$

By definition $4$ ,

The notation $\underset{n\to \infty }{\lim }{n}^3=\infty$ means that for every positive number $M$ there is an integer such that.

In case $n>N$ then $a_n>M$ .

Suppose that $M$ be an positive integer such that

  $\begin{array}{l}{n}^3>M\\ n>\sqrt[3]{M}\\ n^3>M\\ a_n<M\end{array}$

Thus, from the definition $4$ result is $\underset{n\to \infty }{\lim }{n}^3=\infty$ .