Absolute and Conditional Convergence Problem

[atarr3]

Homework Statement

Test the series for (a) absolute convergence, and (b) conditional convergence.

$$\sum\left(-1\right)^{k+1}\frac{k^{k}}{k!}$$

Homework Equations

The Attempt at a Solution

So I tried taking the absolute value and then applying the ratio test, which, after simplifying gives me $$\frac{\left(k+1\right)^{k}}{k^{k}}$$ and then using the root test, but that simplifies to $$\frac{k+1}{k}$$ which converges at 1 and therefore those tests are inconclusive.

 

[VeeEight]

Is k^k > k! true for all k?

 

[atarr3]

Ahhhhh... divergence test. Wow that was a lot easier than I thought it was. So the term goes to infinity and the series diverges. Thank you so much for your help!