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Math Help - Real Analysis NEED HELP QUICK

  1. #1
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    Real Analysis NEED HELP QUICK

    need help with this problem

    show that:

    lim as n goes to infinity of a^n/n!=0 for all a in R
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by economist View Post
    need help with this problem

    show that:

    lim as n goes to infinity of a^n/n!=0 for all a in R
    There is a theorem that states that if \sum_{n=1}^{\infty}a_n converges, then \lim_{n\to\infty}a_n = 0.

    Looking at the sum \sum_{n=1}^{\infty}\frac{a^n}{n!}, use the ratio test. \lim_{n\to\infty}\frac{\frac{a^{n+1}}{(n+1)!}}{\fr  ac{a^n}{n!}} = \lim_{n\to\infty}\frac{a}{n+1} = 0

    Thus you sum converges, and by the theorem \lim_{n\to\infty}\frac{a^n}{n!}=0
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