Proving convergence of infinite series

[utleysthrow]

Homework Statement

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

Does this series converge as n -> infinity?

Homework Equations

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).

 

[foxjwill]

Your proof appears to be valid.