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 easy a_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 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 :) |

euclidean | I know.. :) but consider you wanna go back to standard power series or 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 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? |

futurist | when does lim(f(g(x)) = f(lim(g(x))) ? i'm having trouble finding the rule.. |