# Series - Testing for Convergence / Divergence

by steelphantom
Tags: convergence, divergence, series, testing
 P: 159 I have a few series which I'm having trouble proving whether they converge or diverge. I know the following tests for convergence: comparison test, ratio test, n-th term test, and root test. Here are the series and what I have tried so far: $$\sum$$ n -1 / n2 : I'm assuming this series diverges, since it behaves like 1/n, which also diverges. I'm trying to use the comparison test to see if I can find a "smaller" series which also diverges, but coming up blank. I tried the ratio test to no avail, since it gives 1. $$\sum_{n=2}^\infty$$ 1 / (n + (-1)n)2 : I'm really not sure where to begin with this one. The (-1)n is really throwing me off. I'm assuming this converges. And finally, $$\sum$$ n! / nn : I tried the ratio test, canceling out the factorial and getting the ratio of nn / (n + 1)n. This limit seems to be 1, so the ratio test doesn't really help me here. Any suggestions? Thanks for any help!