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Math Help - Definite Integral

  1. #1
    Junior Member
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    Definite Integral

    \sum_{i=1}^n \frac{x^i}{i}
    how to change this summation into a definite integral ???

    \sum_{i=1}^n x^{i-1} = \frac{1}{1-x}
    integrate both sides, LHS will have \sum_{i=1}^n \frac{x^i}{i}
    but what is the definite limit to take???

    thanks
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by graticcio View Post

    \sum_{i=1}^n x^{i-1} = \frac{1}{1-x}
    This identity is wrong, to see this look at the left hand side, this is finite
    for all x, now look at the right hand side this diverges near x=1.

    What you have is a finite geometric series and:

    \sum_{i=1}^n x^{i-1} = \frac{1-x^n}{1-x}

    Now consider:

    \int_0^x \left[ \sum_{i=1}^n \zeta^{i-1} \right] d \zeta = \int_0^x \frac{1-\zeta^n}{1-\zeta} d \zeta

    RonL
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