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Math Help - Evaluate function?

  1. #1
    Super Member fardeen_gen's Avatar
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    Evaluate function?

    If f(x) = \frac{a^x}{a^x + \sqrt{a}}\ (a > 0), evaluate \sum_{r = 1}^{2n - 1} 2f\left(\frac{r}{2n}\right)
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    Quote Originally Posted by fardeen_gen View Post
    If f(x) = \frac{a^x}{a^x + \sqrt{a}}\ (a > 0), evaluate \sum_{r = 1}^{2n - 1} 2f\left(\frac{r}{2n}\right)
    And according to your previous post, this should be equal to (2n-1)
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    Quote Originally Posted by fardeen_gen View Post
    If f(x) = \frac{a^x}{a^x + \sqrt{a}}\ (a > 0), evaluate \sum_{r = 1}^{2n - 1} 2f\left(\frac{r}{2n}\right)
    Write f(x) = \frac1{1 + a^{\frac12 - x}}

    Observe that f\left(\frac{r}{2n}\right) + f\left(\frac{2n - r}{2n}\right)  = 1

    Thus grouping terms you get the sum to be 2n-1 and not 1.
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