Determine the convergence/divergence of this series with all possible tests

1. Jul 23, 2012

mathnoobie

1. The problem statement, all variables and given/known data
series from n=0 to infinity of
(n+1)/3^n

Did I do this correctly? Also, I'm pretty positive that I can use the comparison test/limit comparison test, but I can't seem to figure out what to compare it to, could someone shed a little light? For the integral test, the integral is quite intimidating and I don't really know where to start, could someone suggest what integration technique to try?

relevant equations:
Ratio Test

3. The attempt at a solution
so first I used the ratio test
lim n->infinity |((n+2)/3^(n+1)) * ((3^n)/(n+1))|
= lim n->infinity (n+2)/(3(n+1))
dividing by n/n
you have the limit = 1/3<1, therefore it converges by the ratio test.

2. Jul 23, 2012

micromass

Staff Emeritus
That's good.

If you want to do the comparison test, then maybe apply

$$(n+1)<2^n$$

for large n?

3. Jul 23, 2012

mathnoobie

So then I would have a convergent geometric series since 2/3<1.
Wow, beautiful. I would've never thought of that on my own, been staring at it for 30minutes trying to compare it to random things.

4. Jul 24, 2012

johnqwertyful

You could simply divide and you would have two geometric series.

Also trivially, it's probably 1 to infinity, because of the n^3 on the bottom.

5. Jul 24, 2012

mathnoobie

I'm confused on where the n^3 came from. Was that a typo and you meant 3^n?

6. Jul 24, 2012