Convergence of Series with Variable Exponent and Negative Power

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The discussion focuses on determining the convergence or divergence of the series Sum(n=1 to infinity) ((n+1)^(n-1))/(-n)^n. Initial attempts using the Ratio and Root tests yielded inconclusive results. Participants suggest applying the alternating series test and emphasize the importance of evaluating the limit and showing that the series is decreasing. One user recommends breaking the series into manageable components to identify a finite limit. The conversation highlights the challenges in proving convergence and the need for further analysis.
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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 :biggrin:

Mike
 
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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.
 
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
 
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?
 

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