Textbook Problem

1. (a) Suppose that Σ a_n and Σ b_n are series with positive terms and Σ b_n is divergent. Prove that if

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

then Σ a_n is also divergent.

1. (b) Use part (a) to show that the series diverges.
   1. (i) ${\displaystyle \underset{n=2}{\overset{\infty }{\sum }}\frac{1}{\ln n}}$
   2. (ii) ${\displaystyle \underset{n=1}{\overset{\infty }{\sum }}\frac{\ln n}{n}}$