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Infinite power series

  1. Aug 19, 2008 #1
    1. The problem statement, all variables and given/known data

    Show that if a function f(x) can be expressed as an infinite power series, then it has the form

    f(x) = f(x0) + [tex]\sum^{\infty}_{n = 1}[/tex][tex]\frac{f^{n}(x0)}{n!}[/tex][tex](x - x0)^{}[/tex]

    2. Relevant equations

    3. The attempt at a solution

    I know that for an infinite power series:

    = f(a) + [tex]\frac{f'(a)}{1!}[/tex](x - a) + [tex]\frac{f''(a)}{2!}[/tex][tex](x - a)^{2}[/tex]....

    which can be simplified into the above expression. But is there any groundwork that the question asks to get to this point here? Im thinking for 6 marsk i cant just right down the two lines...
  2. jcsd
  3. Aug 19, 2008 #2


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    Science Advisor
    Homework Helper

    A power series has the form [itex]
    f(x)= \sum^{\infty}_{n = 0}
    a_n (x-a)^n[/itex]. You want to show [itex]f_n(a)/n!=a_n[/itex]. Differentiate the power series n times and put x=a.
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