Series converge/Diverge?

1. Jan 19, 2010

mikethemike

1. The problem statement, all variables and given/known data

Decide on the convergance/divergance of the following series:

Sum(n=1 to n= infinity) ((n+1)^(n-1))/(-n)^n

where ^ is to the power of and / is divided by.

2. The attempt at a solution

I've used both the Ratio and Root test which are inconclusive (ie. R=1, K=1). Tried changing it around to fit the Leibiniz criterion (and failed). I'm not sure where to go from here...

Thanks for any help

Mike

2. Jan 19, 2010

Dick

Don't give up on the alternating series test. Take the absolute value. Can you show the limit is 0? Now try to show it's decreasing by showing the derivative of the log is negative.

3. Jan 20, 2010

mikethemike

Hey Dick,

I'm having trouble convincing myself that the limit is in fact, zero. I can't seem to prove this, even with the absolute value. Can you point me in the right direction?

Thanks

Mike

4. Jan 20, 2010

Dick

Sure. Break it into (n+1)^(n-1)/n^(n-1) times 1/n. The first factor has a finite limit. Can you find it?