Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Remainder estimate for product series

  1. Aug 7, 2011 #1
    Suppose I have two series

    [itex]A=\sum_{n=0}^\infty a_n[/itex]

    [itex]B=\sum_{n=0}^\infty b_n[/itex]

    and I have estimates for the remainders of each one:

    [itex]\sum_{n=N}^\infty a_n \le R^N_A[/itex]

    [itex]\sum_{n=N}^\infty b_n \le R^N_B[/itex]

    Consider the product series

    [itex]AB=\sum_{n=0}^\infty c_n[/itex]

    where [itex]c_n=\sum_{i=0}^n a_i b_{n-i}[/itex]. Is it possible to derive an estimate for the remainder of [itex]C[/itex] based on the ones for [itex]A[/itex] and [itex]B[/itex]?
     
  2. jcsd
  3. Aug 7, 2011 #2

    I like Serena

    User Avatar
    Homework Helper

    Hi bruno67! :smile:

    Let's define [itex]A_N = \sum\limits_{n=0}^N a_n[/itex] and [itex]B_N = \sum\limits_{n=0}^N b_n[/itex].

    After writing out your formulas, I found I can write your remainder for C as:
    [tex]R_C^N = A_{N-1} R_B^N + B_{N-1} R_A^N[/tex]

    Is that what you're looking for?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Remainder estimate for product series
  1. Product Series (Replies: 12)

  2. Product of a serie (Replies: 6)

Loading...