I'm attempting to work with generating functions that are relatively simple to work with in what I believe is "closed form"; that is, the equation that represents the function. However, the coefficients are relatively complex and difficult to work with.
My problem is that I'd like to find the limit as the index of the coefficient approaches infinity, or one coefficient that is very deep into the equation. In other words, I'd like to find like the millionth term, or I'd also be even happier with the term that represents infinity. However, like I said, I'm having trouble extracting the individual coefficients, and it may be almost impossible to do so.
So I wonder if there is a method or methods that can extract one coefficient. My guess is that I might be able to use the limit to do so.Example
For the generating function
the inifinite term approaches
Please help me find the infinite or near infinite terms!
Can you give the specific function you're working with whose coefficients are "complex and difficult to work with"?
What are you trying to find the limit of
as your index approaches infinity?
You do realize that convergence of the function isn't really necessary for most of the things you care about, right? Furthermore if your function does
converge then the nth term HAS
to go to zero by the nth term test for convergence.
Lastly, if you're dealing with "the generating function for f(x)" then the millionth term is just f(1.000.000).
If you're still having trouble, please say so, and please be specific so that it'll be easier to help you.