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Math Help - Sum of Power Series

  1. #1
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    Sum of Power Series

    Compute \sum_{0}^{\infty}(n^{2}+n)x^{n} for fixed values of x.

    I know this sum is infinity when \left|x \right|\geq1, but how do you compute the value of the sum explicitly when \left|x \right|<1? The best I can seem to do is to use the Gaussian formula for the sum of the first n integers to deal with the (n^{2}+n) term. Any suggestions?
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  2. #2
    Senior Member apcalculus's Avatar
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    Quote Originally Posted by joeyjoejoe View Post
    Compute \sum_{0}^{\infty}(n^{2}+n)x^{n} for fixed values of x.

    I know this sum is infinity when \left|x \right|\geq1, but how do you compute the value of the sum explicitly when \left|x \right|<1? The best I can seem to do is to use the Gaussian formula for the sum of the first n integers to deal with the (n^{2}+n) term. Any suggestions?
    Integration by parts (integral test) test to find the two sums separately?

    \sum_{0}^{\infty}(n^{2}x^{n})

    and

    \sum_{0}^{\infty}(nx^{n})<br />

    In the first sum you would have two use Parts twice. I hope this helps.
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  3. #3
    Moo
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    Hello,

    (n^2+n)x^n=n(n+1) x^n

    Now rewrite this as \frac 1x \cdot n(n+1) x^{n-1}

    And consider differentiating twice the power series \frac{x}{1-x}=\sum_{n\geq 0} x^{n+1}
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