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

Watch
username4136150
Badges: 16
Rep:
?
#1
Report Thread starter 2 years ago
#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!
0
reply
Gregorius
Badges: 14
Rep:
?
#2
Report 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
reply
RichE
Badges: 15
Rep:
?
#3
Report 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.
0
reply
username4136150
Badges: 16
Rep:
?
#4
Report Thread starter 2 years ago
#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??
0
reply
username4136150
Badges: 16
Rep:
?
#5
Report Thread starter 2 years ago
#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?
0
reply
RichE
Badges: 15
Rep:
?
#6
Report 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.)
0
reply
X

Quick Reply

Attached files
Write a reply...
Reply
new posts
Back
to top
Latest
My Feed

See more of what you like on
The Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

Personalise

Do you think receiving Teacher Assessed Grades will impact your future?

I'm worried it will negatively impact me getting into university/college (97)
39.43%
I'm worried that I’m not academically prepared for the next stage in my educational journey (27)
10.98%
I'm worried it will impact my future career (18)
7.32%
I'm worried that my grades will be seen as ‘lesser’ because I didn’t take exams (57)
23.17%
I don’t think that receiving these grades will impact my future (30)
12.2%
I think that receiving these grades will affect me in another way (let us know in the discussion!) (17)
6.91%

Watched Threads

View All
Latest
My Feed

See more of what you like on
The Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

Personalise