proving convergence of infinite series

utleysthrow

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

[tex]\sum \frac{(-1)^{n}}{n+n^{2}}[/tex]

Does this series converge as n -> infinity?

2. Relevant equations



3. The attempt at a solution

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



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

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

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

foxjwill

Your proof appears to be valid.