Convergence of Series with Variable Exponent and Negative Power

[mikethemike]

Homework Statement

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

 

[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.

 

[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

 

[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?