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Math Help - sum of a series

  1. #1
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    sum of a series

    It's been awhile since I have done these types of problems.

    Can the sum of a series have a factorial in the answer?

    this is my problem: Summation of (n/(n+1)!) with n>= 1

    while working it out, I found that this is 1/(n+1)(n-1)!

    can that be an answer?
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  2. #2
    Super Member malaygoel's Avatar
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    Quote Originally Posted by Nichelle14
    It's been awhile since I have done these types of problems.

    Can the sum of a series have a factorial in the answer?

    this is my problem: Summation of (n/(n+1)!) with n>= 1

    while working it out, I found that this is 1/(n+1)(n-1)!

    can that be an answer?
    Are you familiar with the exponential series, expansion of e^x?

    KeepSmiling
    Malay
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  3. #3
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    Consider a function,
    <br />
f(x)=\frac{x^2}{2!}+\frac{2x^3}{3!}+\frac{3x^4}{4!  }+\frac{4x^5}{5!}+...<br />
    Its derivative is, (since it is absolutely convergent)
    f'(x)=x+\frac{x^2}{1}+\frac{x^3}{2!}+\frac{x^4}{3!  }+...
    Thus, (since it is abolsulte convegent)
    f'(x)=x\left( 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+... \right)
    Now, (as malaygoel said) are you familiar with,
    e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...
    Then you are left with,
    f'(x)=xe^x
    Thus, to solve this diffrencial equation find,
    \int xe^x dx use integration by parts with,
    u=x then, u'=1 and v'=e^x thus, v=e^x.
    Thus, (by parts)
    xe^x-\int e^xdx =xe^x-e^x+C
    Thus,
    f(x)=xe^x-e^x+C
    we see that, f(0)=0
    Thus,
    0e^0-e^0+C=0 thus, C=1
    Finally our required function is,
    f(x)=xe^x-e^x+1
    Note, the infinite sum, as you said,
    \frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+...
    Is, the same as f(1) which in this case is,
    1e^1-e^1+1=1
    Last edited by ThePerfectHacker; June 25th 2006 at 05:28 PM.
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  4. #4
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    I vaguely remember the exponential series of e^x.

    I was able to follow your work though.

    This may be a silly question but what exactly is the answer? Do I need to do more?

    Also, when you were trying to find C, why did you set the function equal to 1?
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  5. #5
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    Quote Originally Posted by Nichelle14
    I vaguely remember the exponential series of e^x.

    I was able to follow your work though.

    This may be a silly question but what exactly is the answer?
    2, because that series as I explained is f(1)=1.

    Also, when you were trying to find C, why did you set the function equal to 1?
    My error. You are right.
    I editted my post.
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