# Is the factorial function decreasing?

• Apr 20th 2010, 01:28 PM
DBA
Is the factorial function decreasing?
I need to know if the factorial function is decreasing.
This is the second condition for the Alternating Series Test.

The Alternating Series:

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

I need to show that

$\displaystyle f(x) = \frac{x!}{(2x-1)!}$

decreasing.

Normally, I just take the first derivative but since I have a factorial I don't know.
Can someone let me know how I have to do that?

Thanks.
• Apr 20th 2010, 01:35 PM
TKHunny
Factorials have a symbiotic relationship with the Ratio Test.

Divide any term by the term before it and see how it goes.

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

Simplify that. >1, increasing, <1, decreasing
• Apr 20th 2010, 01:56 PM
DBA

Does it mean that I do not use the Alternating Series Test at all?
Do I just use the Ratio Test and take the limit from the Ratio or do I use the ratio just to determine if it is decreasing?

And how do I find the ratio?

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

I do not know how to simplify the last term.
Can I write

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