1. ## cauchy

given that {a_n} is a sequence of numbers that are greater than 0 and that sum a_n diverges. show that sum a_n / (1+ a_n) diverges.

how do i show that?

i know that a_n / (1+ a_n) = 1- 1/ (1+ a_n) but what else?

2. We can break the problem into two cases.

Case 1: Suppose that $\displaystyle a_n\not \to 0$. Then,

$\displaystyle \displaystyle \sum \frac{a_n}{1+a_n}=\sum \left(1-\frac{1}{1+a_n}\right)$

But then the terms of the sum on the right do not approach zero either, so the sum is divergent.

Case 2: Suppose that $\displaystyle a_n\not \to 0$. Try using the limit comparison test.