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Math Help - sum of convergent series

  1. #1
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    sum of convergent series

    Evaluate the sum of the convergent series [(-1)^n]/[(n+1)2^(2n+5)] as n goes from 0 to infinity.
    Hint: Consider the function f(x) = (x^3)log(1+x^2)
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  2. #2
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    Yes, consider it! What is the MacLaurin series of that function? (You don't have to take derivatives, etc. to find that. The MacLaurin series for log(1+ x) is x- x^2/2+ x^3/3+ \cdot\cdot\cdot= \sum_{n=1}^\infty (-1)^nx^n/n. Replace that "x" with " x^2" and multiply each term by x^3.) What should you set x equal to to get the series you want?
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  3. #3
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    ooh....Why thank-you for clarifying that.
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  4. #4
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    hold on. Then it becomes the sum of (-1)^(n-1) [(x^5)^n]/n as n goes from 1 to infinity. In this case, the series matches the entry for ln(1+x) with x = x^5. So doesn't the sum = ln(1+x^5)? But that's weird. What do you mean by setting x equal to some value?
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