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?
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.