1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Multiplication of power series

  1. Apr 19, 2010 #1
    1. The problem statement, all variables and given/known data

    Suppose that the power series [tex]\Sigma[/tex][a]_{n}[/tex]z^{n} and
    \Sigma b_{n} z^{n} havr radii of convergence R! and R2 respictively. Prove that the radius
    of convergence of the multiplication is at least R1 * R2
    2. Relevant equations

    3. The attempt at a solution

    I found that the convergence radius is > min(R1, R2)
  2. jcsd
  3. Apr 19, 2010 #2
    Assuming you are talking about the Cauchy product of power series, namely [tex]\sum c_n z^n[/tex] where [tex]c_n = \sum_{0\leq j\leq n} a_j b_{n-j}[/tex], then you are correct (but the sign should be [tex]\geq[/tex] rather than [tex]>[/tex]), and the problem statement is wrong. For instance, if [tex]a_n = b_n = 2^{-n}[/tex] so that [tex]R_1 = R_2 = 2[/tex], then [tex]c_n = \frac{n + 1}{2^n}[/tex], and [tex]\limsup_{n\to\infty} \left( \frac{n + 1}{2^n} \right)^{1/n} = \frac12[/tex] so that the radius of convergence of [tex]\sum c_n z^n[/tex] is also [tex]2[/tex].

    Perhaps the question intended to refer to a different notion of product, such as [tex]\sum d_n z^n[/tex] where [tex]d_n = a_n b_n[/tex]?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook