Chapter 9.4, Problem 66E

(a)

To determine

To Prove:

If $\underset{n\to \infty }{\lim }\frac{a_n}{b_n}=\infty ,{\displaystyle \sum a_n}$ and ${\displaystyle \sum b_n}$ diverges then the series ${\displaystyle \sum _{n=1}^{\infty }{\left(n+2\right)}^2}$ will also diverge.

(b)

To determine

To Prove:

If $\underset{n\to \infty }{\lim }\frac{a_n}{b_n}=\infty ,{\displaystyle \sum a_n}$ and ${\displaystyle \sum b_n}$ diverges then the series ${\displaystyle \sum _{n=1}^{\infty }\frac{\ln }{n}}$ diverges.

(c)

To determine

To Prove:

If $\underset{n\to \infty }{\lim }\frac{a_n}{b_n}=\infty ,{\displaystyle \sum a_n}$ and ${\displaystyle \sum b_n}$ diverges then the series ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{\ln n}}$ diverges.

(d)

To determine

To Prove:

If $\underset{n\to \infty }{\lim }\frac{a_n}{b_n}=\infty ,{\displaystyle \sum a_n}$ and ${\displaystyle \sum b_n}$ diverges then the series ${\displaystyle \sum _{n=1}^{\infty }\frac{e^n}{\sqrt{n}}}$ diverges.