Chapter 9.3, Problem 46E

Solution

To identify: Which of the following sequences is divergent.

Options (a) $\left(\frac{n+100}{n}\right)$

Options (b) $\sqrt{100}$

Options (c) ${\pi }^{-n}$

Options (d) $\left(\frac{2n+2}{n+1}\right)$

Options (e) $n^{-2}$

Option (b) is correct option $\sqrt{100}$ sequences is divergent.

Given information:

The given is sequences is divergent

Explanation:

Consider the given information

A sequence diverges when the limit is infinite or nonexistent.

Choice a Converges to $1$ since $\underset{n\to 8}{\lim }\frac{n+100}{n}=\frac{1}{1}=1$ 1. (ratio of the leading coefficients since the degrees of the numerator and denominator are equal)

Choice (b) diverges since $\underset{n\to 8}{\lim }\sqrt{n}=\infty$

Choice (c) converges to $0$ since $\underset{n\to 8}{\lim }{\pi }^{-n}=\underset{n\to 8}{\lim }\frac{1}{{\pi }^n}=0$ .

Choice (d) converges to $2$ since $\underset{n\to 8}{\lim }\frac{2n+2}{n+1}=\frac{2}{1}=2$ . (ratio of the leading coefficients since the degrees of the numerator and denominator are equal)

Choice (e) converges to 0 since $\underset{n\to 8}{\lim }{\text{ n}}^{-2}=\underset{n\to 8}{\lim }\frac{1}{n^2}=0$

So, the is correct option (b).