# Proving convergence of infinite series

1. Oct 22, 2009

### utleysthrow

1. The problem statement, all variables and given/known data

$$\sum \frac{(-1)^{n}}{n+n^{2}}$$

Does this series converge as n -> infinity?

2. Relevant equations

3. The attempt at a solution

First, by the absolute convergence test, $$\sum \frac{(-1)^{n}}{n+n^{2}}$$ should converge if $$\sum \left|\frac{(-1)^{n}}{n+n^{2}}\right|$$ converges.

Second, $$\sum \left|\frac{(-1)^{n}}{n+n^{2}}\right| = \frac{1}{n+n^{2}}< \sum 1/n^{2}$$

Because the sum 1/n^2 converges, by the comparison test, $$\sum \left|\frac{(-1)^{n}}{n+n^{2}}\right|$$ converges.

Which means that $$\sum \frac{(-1)^{n}}{n+n^{2}}$$ converges as well (by the absolute convergence test).

2. Oct 22, 2009

### foxjwill

Your proof appears to be valid.