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Homework Help: Closed form expression for f(x) = sigma (n = 1 to infinity) for x^n / [n(n+1)]

  1. Mar 18, 2010 #1
    1. The problem statement, all variables and given/known data

    Consider the power series.

    sigma (n=1 to infinity) x^n / [n(n+1)]

    if f(x) = sigma x^n / [n(n+1)], then compute a closed-form expression for f(x).

    It says: "Hint: let g(x) = x * f(x) and compute g''(x). Integrate this twice to get back to g(x) and hence derive f(x)".

    2. Relevant equations

    None? Well, see number 3.

    3. The attempt at a solution

    OK so i calculated the interval of convergence to be

    -1 <= x <=1

    and i'm following the hint so.

    g(x) = x f( x)
    and f(x) = x/2 + x²/6 + x^3 / 12 + x^4 / 20 ...

    and x f(x) = x²/2 + x^3 /6 + x^4 / 12 ...

    and g(x) = above.

    so g'(x) = x + x²/2 + x^3 / 3 + x^4 / 4...

    and g''(x) = 1 + x + x² + x^3 + x^4

    and I recognized g''(x) to be the power series at x = 0 for

    1 / (1 - x)

    so i set g'' (x) = 1 / (1-x)

    and then the hint said to integrate g''(x) twice, so

    g'(x) = - ln (1-x).

    now i'm stuck. How can I integrate -ln (1-x) again? or am i supposed to use what i know (i.e. f(1) = 1 and f(-1) = -1 - ln2?) (that is, i calculated the sum of f(x) if x = 1 and x = -1.

    THANK YOU!!!
     
  2. jcsd
  3. Mar 18, 2010 #2

    CompuChip

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    Homework Helper

    What is the derivative of x ln(x) - x?
     
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