Generating functions and summation

[Pere Callahan]

Hello this is something that just crossed my mind:

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

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

 

[rochfor1]

No. w_1=1 and w_2=0 shows it's impossible, as $$w_n = w_1^n$$.