#math - Sun 1 Apr 2007 between 04:23 and 04:33

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flamingspinachanother way to define the infinite product is: \prod_{i=0}^\infty a_n = exp(\sum_{i=0}^\infty \log |a_n|)
TRWBW: okay, that's fine :o
euclideanI'm having trouble understanding generating functions. More precisely, consider you have a sequence given by its recurrence equation, {A_n}. Then you create a power series Sum[A_n x^n, n = 0, infinity]. How can you tell if that series will converge or not since you don't know the general term of the sequence {A_n}?
futuristflamingspinach right
flamingspinacheuclidean: you don't assume that it does. it's a formal power series which need not converge outside of x=0
euclideanflamingspinach: right, so consider I get this formula using formal series: f(x) = Sum[(n^2 + 1) * x^n, n=0,infinity]. How do I get the general term of {A_n} out of this result?
flamingspinach: the series should converge in the end right?
flamingspinacher - you have the general term already, that's easy
a_n = n^2 + 1
euclideanI'm trying to figure out how they make this move from formal power series back to standard power series
flamingspinachthere's no need to, really o_O
euclideando I have to prove that using induction or something?
flamingspinachthe generating function is just a tool to keep track of your sequence and relate it to a power series
uh, no - a_n is defined as the coefficient of x^n
so just look at your series, it's sitting right there already
:)
euclideanI know.. :) but consider you wanna go back to standard power series
or consider you work with standard power series from the first place
flamingspinachwhat do you mean "standard" power series?
euclideanwhen do you test for convergence?
flamingspinachyou don't
euclideanI mean series in which x is a real number, for example
flamingspinachit always converges somewhere, how does it matter?
x=0 will always converge
euclideanit needs to converge for x =/= 0
flamingspinachwhy?
euclideanI am using Maclaurin series.. to get that formula a_n = d^n f/d x^n (0) * 1/n!
where f is the sum of the power series. but this only make sense when the series converges for x =/= 0
flamingspinachI'm not sure I quite understand what your question is...
euclideanmy question is when should I test for convergence?
I certainly cannot test from the start, since I don't know the general term
flamingspinachI don't understand why you need to
euclideanis it ok to suppose the series is convergent for x =/= 0?
flamingspinachthough I'm not exactly an expert on generating functions - maybe TRWBW can take a look? :P
euclideanbecause basically I'm not using generating functions, just plain old power series
TRWBW: do you have any idea how I should do this?
TRWBW: I'm trying to avoid generating functions and use real power series from the start
TRWBW: that's because I need the Maclaurin formula that relates the derivatives of the sum f of the power series, to the coefficients of the series
TRWBW: so the power series needs to converge for x =/= 0 right?
futuristwhen does lim(f(g(x)) = f(lim(g(x))) ? i'm having trouble finding the rule..

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