Calculus...infinite series?

Question

This question is about alternating series. I understand that an alternating series converges if the limit tending to infinity is 0, and if the terms are decreasing throughout. If for a series, one of these tests for convergence fails and the alternating series test fails, what tests can you use to conclude that the series diverges? I'm thinking nth term, p-series, geometric series, but i'm not sure...please help =]. A sample question that I am working on is...

the series (-1)^n 3^(n-2) / 2^n with n from 1 to infinity
p.s. there is a multiplication between the n and the 3

Answer 1

The limit of the nth term as n --> infinity is
lim 3^(n-2)/2^n
=lim [3^n/(9*2^n)) which clearly goes to + infinity as n --> infinity. Thus the series diverges.

The series converges if |An| < |A(n-1)| and
lim An = 0
n --> infinity

That's all there is to it. Why look for other ways to show convergence/divergence?

Answer 2

The series (-1)^n 3^(n-2) / 2^n with n from 1 to infinity diverges because r = 3/2 > 1.