1. Alternate Series Test

((-1)^(n+1)*n^(2)*3^n)/2^n

can anyone help me with this problem. i tried the alternating series test and it came out inconclusive, but i don't know what other test to try.

2. lim (n^2)*(3^n)/2^n = lim n^2 (3/2)^n = inf

Therefore the series diverges

3. whats the condition to prove that. just because the series equals 0, then it diverges?

4. 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

5. Originally Posted by needhelp101
whats the condition to prove that. just because the series equals 0, then it diverges?
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.

6. Originally Posted by needhelp101
((-1)^(n+1)*n^(2)*3^n)/2^n

can anyone help me with this problem. i tried the alternating series test and it came out inconclusive, but i don't know what other test to try.
$a_n = (-1)^{n+1}n^2\left(\frac{3}{2}\right)^n$

This is unbounded and therefore $\sum_{n=1}^{\infty}a_n$ diverges.