I have SUM [ ((-1)^n * n!)/n^n ] from n=1 to +inf . I need to check both absolute and conditional convergence.

Now this is very tricky.. the (-1)^n tells me i need to try with the alternating series test.. Than An= n!/n^n , and lim n->inf An is impossible to find for me.

When i tried with Ratio test for absolute convergence i got 1 as result which tells me nothing .

My question here is: Can i cancel n! , and say the series will converge is the series SUM [ 1/n^n ] from n=1 to +inf is converging too ? The last series can be considered as p-series or a geometric series , and it converges since n will always be positive .

At the end does that tells me that the first series is also absolute convergent .