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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]?
     
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