Classifying an Alternating Factorial Series

[dba]

Homework Statement

Classify the series as absolute convergent, conditionally convergent, or divergent.

$$\Sigma^{\infty}_{k = 1} (-1)^{k-1}\frac{k!}{(2k-1)!}$$

Homework Equations

The Alternating Series Test: conditions for convergence
decreasing
lim --> infinity ak = 0

The Attempt at a Solution

I am not sure how to find out if the series is decreasing. Since it is a factorial, I cannot take the first deriative test.

I was wondering if I should use the Alternating Series Test at all, since it is an factorial.

Can someone help me here?

 

[lanedance]

try the alternating series test

a ratio of terms will show if the magnitude of terms decrease monotonically

 

[dba]

Thanks.

If I try the ratio I got stuck with the factorial.

$$\frac{(k+1)!}{(2(k+1)-1)!} * \frac{(2k-1)!}{k!} = \frac{(2k-1)!}{(2(k+1)-1)!} * (k+1)$$


Can I write

$$(2(k+1)-1)! = (2k+2-1)! = (2k+1)!$$

 

[HallsofIvy]

Thanks.

If I try the ratio I got stuck with the factorial.

$$\frac{(k+1)!}{(2(k+1)-1)!} * \frac{(2k-1)!}{k!} = \frac{(2k-1)!}{(2(k+1)-1)!} * (k+1)$$


Can I write

$$(2(k+1)-1)! = (2k+2-1)! = (2k+1)!$$

Yes, of course. And so you have (2k-1)!/(2k+1)!= (2k-1)!/[(2k+1)(2k)(2k-1)!]

 

[dba]

Oh, ok.
Thank you! I was able to solve this one