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Math Help - How to expand factorials in ratio test

  1. #1
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    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.
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  2. #2
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    Re: How to expand factorials in ratio test

    2n!=2n x (2n-1) x (2n-2) x ...
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    Re: How to expand factorials in ratio test

    Quote Originally Posted by pickslides View Post
    2n!=2n x (2n-1) x (2n-2) x ...
    I assume you mean (2n)!
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    Re: How to expand factorials in ratio test

    Thanks Hayden, I (did)!
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    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)
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    Re: How to expand factorials in ratio test

    Looks good to me.
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    Re: How to expand factorials in ratio test

    Quote Originally Posted by itpro View Post
    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)!
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    Re: How to expand factorials in ratio test

    This problem is so difficult, I still wait for the experts to answer this question.
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    Re: How to expand factorials in ratio test

    Quote Originally Posted by felixjsc View Post
    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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