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.