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Math Help - deduce that limit equals 0

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    deduce that limit equals 0

    Deduce that lim as n goes to infinity of x^n/(n!) equals o for all x
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    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by twilightstr View Post
    Deduce that lim as n goes to infinity of x^n/(n!) equals o for all x
    If \lim\frac{s_{n+1}}{s_n}=L where L<1, then \lim s_n=0

    So, let s_n=\frac{x^n}{n!}. As a result, s_{n+1}=\frac{x^{n+1}}{\left(n+1\right)!}

    Therefore, \lim\frac{s_{n+1}}{s_n}=\lim\frac{x^{n+1}}{\left(n  +1\right)!}\cdot\frac{n!}{x^n}=\lim\frac{x}{n+1}=0  \quad\forall\,x\in\mathbb{R}

    Since 0<1, we can conclude that \lim\frac{x^n}{n!}=0\quad\forall\,x\in\mathbb{R}.

    (Note that \lim is analogous with \lim_{n\to\infty})
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