Chapter 8.3, Problem 42E

(a) Suppose that ${\displaystyle \sum a_n}$ and ${\displaystyle \sum b_n}$ are series with positive terms and ${\displaystyle \sum b_n}$ is convergent. Prove that if

$\underset{n\to \infty }{\lim }\frac{a_n}{b_n}=0$

then ${\displaystyle \sum a_n}$ is also convergent.

(b) Use part (a) to show that the series converges.

(i) ${\displaystyle \underset{n=1}{\overset{\infty }{\sum }}\frac{\ln n}{n^3}}$

(ii) ${\displaystyle \underset{n=1}{\overset{\infty }{\sum }}\frac{\ln n}{\sqrt{n}{e}^n}}$