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Generating functions and summation

  1. Feb 9, 2010 #1
    Hello this is something that just crossed my mind:

    For every real sequence [itex](a_n)_{n\geq 1}[/itex] we can define the generating function
    [tex]
    A(z)=\sum_{n=1}^\infty{a_nz^n}.
    [/tex]
    and this definition suggests that we can compute the sum of the sequence by evaluating A at 1:
    [tex]
    A(1)=\sum_{n=1}^\infty{a_n}
    [/tex]
    provided the sum converges.

    This made me wonder if for any sequence of weights [itex](w_n)_{n\geq 1}[/itex] there is a real number x such that we can compute the sum
    [tex]
    \sum_{n=1}^\infty{w_na_n}
    [/tex]
    as A(x).

    I haven't spent much time on this, but I would be very interested in any thoughts on the topic.

    Thanks,
    Pere
     
  2. jcsd
  3. Feb 9, 2010 #2
    No. w_1=1 and w_2=0 shows it's impossible, as [tex]w_n = w_1^n[/tex].
     
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