# Thread: How to expand factorials in ratio test

1. ## How to expand factorials in ratio test

I have a set of convergence problems that I know I should solve with a ratio test,
but in denominator I have either $2n!$ or $(2n+2)!$ or even $(n+1)!$ and I am little bit shakey with expanding these factorials for cancellation. I am only familiar with 5! or n! but not with factorial of sums or factorials of sums of products of variables and integers.

Is there a rule or a formula how could I do so? Would someone please show me how is and why $(2n+2)!$ expanded.

Thank you.

2. ## Re: How to expand factorials in ratio test

2n!=2n x (2n-1) x (2n-2) x ...

3. ## Re: How to expand factorials in ratio test

Originally Posted by pickslides
2n!=2n x (2n-1) x (2n-2) x ...
I assume you mean (2n)!

4. ## Re: How to expand factorials in ratio test

Thanks Hayden, I (did)!

5. ## Re: How to expand factorials in ratio test

So, for the ratio test when I have $\frac{(2(n+1))!}{2n!}$ be:

$\frac{(2n+2)(2n+1)(2n)!}{(2n)!}$ simplifying to $(2n+2)(2n+1)$

6. ## Re: How to expand factorials in ratio test

Looks good to me.

7. ## Re: How to expand factorials in ratio test

Originally Posted by itpro
So, for the ratio test when I have $\frac{(2(n+1))!}{2n!}$ be:

$\frac{(2n+2)(2n+1)(2n)!}{(2n)!}$ simplifying to $(2n+2)(2n+1)$
If there were brackets around the 2n in the denominator, you'd be correct.

2n! = 2(n!), not (2n)!

8. ## Re: How to expand factorials in ratio test

This problem is so difficult, I still wait for the experts to answer this question.

9. ## Re: How to expand factorials in ratio test

Originally Posted by felixjsc
This problem is so difficult, I still wait for the experts to answer this question.
The experts HAVE answered this question. If there is a more specific question that needs answering, it should have been asked in the first post.

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