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Math Help - about sequences 1

  1. #1
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    about sequences 1

    Hello,

    Show that lim_{n\rightarrow +\infty}n[1-\frac{(n+1)^{n}}{en^{n}}]=\frac{1}{2}.

    Thanks in advance.
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  2. #2
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    Quote Originally Posted by Biscaim View Post
    Hello,

    Show that lim_{n\rightarrow +\infty}n[1-\frac{(n+1)^{n}}{en^{n}}]=\frac{1}{2}.

    Thanks in advance.
    It looks to me like you'll need to develop this expression as a series. See - Wolfram|Alpha
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  3. #3
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    Quote Originally Posted by Biscaim View Post
    Hello,

    Show that lim_{n\rightarrow +\infty}n[1-\frac{(n+1)^{n}}{en^{n}}]=\frac{1}{2}.

    Thanks in advance.
    n[1-\frac{(n+1)^n}{en^n}] = \frac{1 - \frac{1}{e}\Big(1+\frac{1}{n}\Big)^n}{\frac{1}{n}}. By definition of e, the numerator approaches 1 - 1/e * e = 0, so maybe applying L'Hopital's rule will be successful (I haven't tried it).
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