Question

Determine if the series abs converges, conditionally converges, or diverges.

n^n /n!

Answer 1

As \( n\to \infty\)
\[n!>>n^2\]
so the series converges

Answer 2

Using ratio test, I found that it diverges.. But I don't know what test to use to find if it conditionally converges. :/

Answer 3

If it diverges, you don't have to worry about conditional convergence. In general, though, you'd have to show that a given series isn't absolutely convergent. For this, you'd have to show that the following isn't true:
\(\displaystyle\sum |a_n|\text{ converges.}\)

Answer 4

For this series, its always positive. I'm not sure why I used ratio test but I did. I'm confused at when I can use ratio and root tests (only when it's alternating signs?). And I thought a series can converge even if it absolutely diverges

Answer 5

"For this series, it's always positive." Yes.

"I'm not sure why I used the ratio test..." If your series contains a factorial, the ratio test usually works pretty well.

"...confused when to use ratio/root tests (only when alternating signs?)..." Here's a handy chart for knowing when to use which test: http://www.math.tamu.edu/~austin/serieschart.pdf
As for the alternating sign - use the alternating series test. The reason why should be fairly clear.

"I thought a series can converge even if it absolutely diverges." That's not at all the case. If a series diverges, it diverges. It can't converge if it diverges. If a series converges, then it can be either conditionally or absolutely convergent.