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

### NY Lost Funds

 flamingspinach another 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 euclidean I'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}? futurist flamingspinach right flamingspinach euclidean: you don't assume that it does. it's a formal power series which need not converge outside of x=0 euclidean flamingspinach: 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? flamingspinach er - you have the general term already, that's easya_n = n^2 + 1 euclidean I'm trying to figure out how they make this move from formal power series back to standard power series flamingspinach there's no need to, really o_O euclidean do I have to prove that using induction or something? flamingspinach the generating function is just a tool to keep track of your sequence and relate it to a power seriesuh, no - a_n is defined as the coefficient of x^nso just look at your series, it's sitting right there already:) euclidean I know.. :) but consider you wanna go back to standard power seriesor consider you work with standard power series from the first place flamingspinach what do you mean "standard" power series? euclidean when do you test for convergence? flamingspinach you don't euclidean I mean series in which x is a real number, for example flamingspinach it always converges somewhere, how does it matter?x=0 will always converge euclidean it needs to converge for x =/= 0 flamingspinach why? euclidean I 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 flamingspinach I'm not sure I quite understand what your question is... euclidean my question is when should I test for convergence?I certainly cannot test from the start, since I don't know the general term flamingspinach I don't understand why you need to euclidean is it ok to suppose the series is convergent for x =/= 0? flamingspinach though I'm not exactly an expert on generating functions - maybe TRWBW can take a look? :P euclidean because basically I'm not using generating functions, just plain old power seriesTRWBW: 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 startTRWBW: 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 seriesTRWBW: so the power series needs to converge for x =/= 0 right? futurist when does lim(f(g(x)) = f(lim(g(x))) ? i'm having trouble finding the rule..