Show that $\sum(-1)^{n+1}ne^{-n}$ is convergent The Question
Determine if $\sum(-1)^{n+1}ne^{-n}$ converges or diverges
My Work
It's an alternating series, so if it meets its conditions, I could use the alternating series test.
I took the derivative of $\frac{n}{e^n}$ which is $\frac{1-n}{e^n}$ and found it was only greater than $0$ for $n<-1$. I may have made a mistake here, but if I didn't it doesn't look like I can use the alternating series test.
Maybe I could just do a simple test for divergence. But the limit of the $n^{th}$ summand goes to $0$ which unfortunately tells us nothing.
My Question
Did I make any errors in my work that if corrected would lead me to a right solution? Is there another approach I should take? This question is from the section on alternating series in my book, so I think they would want me to use the Alternating Series Test. 

Your derivative is right, and you can conclude that there is decrease from $n=1$ on. You will still have to show that $\lim_{n\to\infty} ne^{-n}=0$ to apply the Alternating Series Test.
I would instead suggest using the Ratio Test to conclude absolute convergence and hence convergence. Less work, and a stronger result.
