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Math Help - Find the sum of this series

  1. #1
    Newbie ZosoPage's Avatar
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    Find the sum of this series

    Find the sum of the following series:

    2+\frac{4}{2!}+\frac{8}{3!}+\frac{16}{4!}+...

    To two decimal places.
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  2. #2
    Behold, the power of SARDINES!
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    Quote Originally Posted by ZosoPage View Post
    Find the sum of the following series:

    2+\frac{4}{2!}+\frac{8}{3!}+\frac{16}{4!}+...

    To two decimal places.
    Remember that

    e^{x}=\sum_{n=0}^{\infty}\frac{x^n}{n!}=1+x+\frac{  x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+...

    so

    e^2=1+2+\frac{4}{2!}+\frac{8}{3!}+\frac{16}{4!}+..  .


    so if you subtract 1 from both sides you get

    e^2-1=2+\frac{4}{2!}+\frac{8}{3!}+\frac{16}{4!}+...
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  3. #3
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    Quote Originally Posted by ZosoPage View Post
    Find the sum of the following series:

    2+\frac{4}{2!}+\frac{8}{3!}+\frac{16}{4!}+...

    To two decimal places.
    e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ...<br />

    e^x - 1 = x + \frac{x^2}{2!} + \frac{x^3}{3!} + ...

    let x = 2
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  4. #4
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    Quote Originally Posted by ZosoPage View Post
    Find the sum of the following series:

    2+\frac{4}{2!}+\frac{8}{3!}+\frac{16}{4!}+...

    To two decimal places.
    This series looks like

    \frac{2}{1!}+\frac{4}{2!}+\frac{8}{3!}+\frac{16}{4  !}+...


    \frac{2^1}{1!}+\frac{2^2}{2!}+\frac{2^3}{3!}+\frac  {2^4}{4!}+...

    \sum_{n=1}^{\infty} \frac{2^n}{n!}

    I hope this helps!
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  5. #5
    Newbie ZosoPage's Avatar
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    So the sum is e^2-1?
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  6. #6
    Behold, the power of SARDINES!
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    Quote Originally Posted by ZosoPage View Post
    So the sum is e^2-1?
    Yes that is the exact sum
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