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Math Help - Converging/Diverging Sequence

  1. #1
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    Question Converging/Diverging Sequence

    Prove whether the following sequence converges or diverges.

    \{\frac{n!}{2^n} \}\limits^\infty_{n=0}

    Since it is a sequence I cannot use any of the tests used for series. Any help would be appreciated.
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  2. #2
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    If n\ge 4 then n!>2^n.
    What should that tell you?
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  3. #3
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    Quote Originally Posted by Plato View Post
    If n\ge 4 then n!>2^n.
    What should that tell you?
    Yeah I know the sequence goes to infinity and that factorials grow much faster than exponentials but is there some other way to prove it? Like taking a limit or is this the only way?
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  4. #4
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    The fact that f(n) > g(n) does not determine whether \lim_{n\to\infty}f(n)/g(n) exists. However, it is easy to show that \frac{1}{2}\cdot\frac{2}{2}\cdot\ldots\cdot \frac{n}{2} becomes arbitrarily large.

    I know the sequence goes to infinity
    This is a synonym for "diverges".
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