Does (-1)^(n+1) * n!/2^n converge absolutely or conditionally?

[thagzone]

Homework Statement

Does this series converge absolutely or conditionally?

Homework Equations

Series from n=1 to ∞ (-1)^(n+1) * n!/2^n

The Attempt at a Solution

In trying to apply the alternating series test, I have found the following:

1.) n!/2^n > 0 for n>0
2.) Next, in testing to see if n!/2^n is decreasing, I found that (n!/2^n)/((n+1)!/2^(n+1)) < 1 for n large.

Stopping here, this suggests the series diverges in its original form. Is this correct? Thank you!

 

[LCKurtz]

You have apparently shown the terms of the series are increasing for n large, so they don't go to zero. So yes, you are correct that the series diverges.