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Math Help - Ratio Test for Convergence

  1. #1
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    Ratio Test for Convergence

    Having a bit of trouble with this one. Can anyone help?
    Many thanks.

    Q. \sum_{n=1}^{\infty}\frac{(n-1)!}{2^n}

    Attempt: U_n = \frac{(n-1)!}{2^n} => u_n_+_1 = \frac{n!}{2^n^+^1}
    \lim_{n \to \infty} |\frac{U_n_+_1}{U_n}| => \lim_{n \to \infty} |\frac{n!}{2^n^+^1}.\frac{2^n}{(n-1)!}| => \lim_{n \to \infty} |\frac{n!}{2^n.2}.\frac{2^n}{(n-1)n!}| => \lim_{n \to \infty}|\frac{1}{2(n-1)}| => \lim_{n \to \infty}|\frac{1}{2n-2}|

    Ans.: (From text book): Series is divergent.
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  2. #2
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    Re: Ratio Test for Convergence

    Quote Originally Posted by GrigOrig99 View Post
    Having a bit of trouble with this one. Can anyone help?
    Many thanks.

    Q. \sum_{n=1}^{\infty}\frac{(n-1)!}{2^n}

    Attempt: U_n = \frac{(n-1)!}{2^n} => u_n_+_1 = \frac{n!}{2^n^+^1}
    \lim_{n \to \infty} |\frac{U_n_+_1}{U_n}| => \lim_{n \to \infty} |\frac{n!}{2^n^+^1}.\frac{2^n}{(n-1)!}| => \lim_{n \to \infty} |\frac{n!}{2^n.2}.\frac{2^n}{(n-1)n!}| => \lim_{n \to \infty}|\frac{1}{2(n-1)}| => \lim_{n \to \infty}|\frac{1}{2n-2}|

    Ans.: (From text book): Series is divergent.
    \left|\frac{n!}{2^n^+^1}.\frac{2^n}{(n-1)!}\right|=\frac{n}{2}
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  3. #3
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    Re: Ratio Test for Convergence

    Great. Thank you.
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