Hello this is something that just crossed my mind:(adsbygoogle = window.adsbygoogle || []).push({});

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

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# Generating functions and summation

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