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Series identity

  1. Nov 27, 2009 #1
    given the infinite power series

    [tex] f(x)= \sum_{n=0}^{\infty}a_ {n}x^{n} [/tex]

    if we know ALL the a(n) is there a straight formula to get the coefficients of the b(n)

    [tex] \frac{1}{f(x)}= \sum_{n=0}^{\infty}b_ {n}x^{n} [/tex]

    for example from the chain rule for 1/x and f(x) could be obtain some combinatorial argument to get the b(n) from the a(n) ??
  2. jcsd
  3. Nov 27, 2009 #2


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    Science Advisor

    Not usually. f(x)=x. There is no power series for 1/f(x).
  4. Nov 28, 2009 #3
    Take the case a_0 not 0. We can obtain the b_n by *long division* of power series.
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