# Alternate Series Test

• Apr 21st 2009, 07:23 PM
needhelp101
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.
• Apr 21st 2009, 08:17 PM
Calculus26
lim (n^2)*(3^n)/2^n = lim n^2 (3/2)^n = inf

Therefore the series diverges
• Apr 21st 2009, 08:54 PM
needhelp101
whats the condition to prove that. just because the series equals 0, then it diverges?
• Apr 21st 2009, 09:00 PM
Calculus26
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
• Apr 21st 2009, 10:10 PM
woof
Quote:

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.
• Apr 21st 2009, 10:28 PM
redsoxfan325
Quote:

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.

$\displaystyle a_n = (-1)^{n+1}n^2\left(\frac{3}{2}\right)^n$

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