Is the factorial function decreasing?

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April 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:

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

I need to show that

$ 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.
April 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.

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

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

Thank you very much for your answer.

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?

$ \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

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

Thanks for your help.
April 21st 2010, 08:21 AM
TKHunny

You have it! Now, keep going until you've no factorial remaining.

Excellent.

For now, the ratio test will tell you increasing or decreasing. Later, it will tell you converging or diverging. Of course, I did not address "=1". That's not quite as useful as not equal to one.

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