lim (n^2)*(3^n)/2^n = lim n^2 (3/2)^n = inf
Therefore the series diverges
The alt series test
For (-1)^n an if an is a decreasing seq
If lim an = 0 then the series converges
If lim an is not 0 (regardless of whether or not it is decreasing) the series diverges this is because the sequence diverges
Remember for any series if the sequence doesn't converge to 0 the series diverges
If you want go to my website where i illustrate with animations alternating sequences
Infinite Sequences
The alt series test does not apply, since that is a test whose conclusion can only tell you about convergence. It's the one sentence above that gives you the justification:
If lim an is not 0 (regardless of whether or not it is decreasing) the series diverges.
The above is true whether the series alternates or not.