# Why do we ignore the issues of convergence in generating functions?

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#1
Hi
I have recently introduced myself to generating functions and, I don't seem to understand it at all. I have a lot of questions to ask but firstly, why do we ignore the issues of convergence?

Thank You!
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2 years ago
#2
(Original post by Quantum Horizon)
Hi
I have recently introduced myself to generating functions and, I don't seem to understand it at all. I have a lot of questions to ask but firstly, why do we ignore the issues of convergence?

Thank You!
If you are talking about probability generating functions, they always converge (in some interval around zero).
0
2 years ago
#3
(Original post by Quantum Horizon)
Hi
I have recently introduced myself to generating functions and, I don't seem to understand it at all. I have a lot of questions to ask but firstly, why do we ignore the issues of convergence?

Thank You!
There is a notion of formal powers series, and generating functions might be seen as such. The ring R[[x]] of formal power series consists of series

a_0 + a_1 x + a_2 x^2 + ...

with coefficients in R which multiply and add as you would expect. One can discuss this ring irrespective of any convergence issues.

In a similar light generating functions might be seen as formal power series, and a convenient way of describing a sequence of numbers in closed form.
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#4
(Original post by RichE)
There is notion of formal powers series, and generating functions might be seen as such. The ring R[[x]] of formal power series consists of series

a_0 + a_1 x + a_2 x^2 + ...

which multiply and add as you would expect. One can discuss this ring irrespective of any convergence issues.

In a similar light generating functions might be seen as a formal power series, and a convenient way of describing a sequence of numbers in closed form.
PRSOM!!

Ah, I see. The existence of closed form of a sequence is only convenient but not necessary. All we want to do is code our sequence into gf, which we can even if the series diverges??
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#5
(Original post by Gregorius)
If you are talking about probability generating functions, they always converge (in some interval around zero).
Thanks for the reply. I was talking about the recurrence relations and gf method of dealing with them.

P.S. Is it true that all Ordinary Generating Functions (or Power Series) converge near origin?
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2 years ago
#6
(Original post by Quantum Horizon)
P.S. Is it true that all Ordinary Generating Functions (or Power Series) converge near origin?
The generating function for the sequence n! would only converge at the origin.

However any sequence which has a generating function in closed form would typically converge around the origin. (Technically if the closed form function is holomorphic at the origin then Taylor's theorem would say it was analytic - expressible as a power series - on an open ball containing the origin.)
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