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Math Help - Ratio test with factorials

  1. #1
    Junior Member
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    Ratio test with factorials

    This problem is tripping me up:

    <br />
\sum_{n=1}^{\infty} \frac {(n!)^2}{(2n)!}<br />

    I'm trying to use the ratio test to see if it converges/diverges and am getting:

    <br />
\lim_{n \rightarrow\ \infty} \frac{((n+1)!)^2}{(2(n+1))!}\times\frac{(2n)!}{(n!  )^2}<br />

    I'm assuming the work is correct so far, and that the ratio test is the right test to use, but I really have no idea where to go from here. I'd like to be able to cancel some stuff but I think I'm lacking a good understanding of how these factorials relate to each other and how to work with them squared. Any help appreciated.
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  2. #2
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    Quote Originally Posted by Shananay View Post
    This problem is tripping me up:

    <br />
\sum_{n=1}^{\infty} \frac {(n!)^2}{(2n)!}<br />

    I'm trying to use the ratio test to see if it converges/diverges and am getting:

    <br />
\lim_{n \rightarrow\ \infty} \frac{((n+1)!)^2}{(2(n+1))!}\times\frac{(2n)!}{(n!  )^2}<br />

    I'm assuming the work is correct so far, and that the ratio test is the right test to use, but I really have no idea where to go from here. I'd like to be able to cancel some stuff but I think I'm lacking a good understanding of how these factorials relate to each other and how to work with them squared. Any help appreciated.
    n!=n(n-1)(n-2)...(3)(2)(1)
    5!=(5)(4)(3)(2)(1)=5(4!)
    ((n+1)!)^2=((n+1)n!)^2=(n+1)^2(n!)^2.
    (2(n+1))!=(2n+2)!=(2n+2)(2n+1)(2n)!.
    Now, You cancel them out.
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