Results 1 to 2 of 2

Math Help - Cauchy Product

  1. #1
    Member
    Joined
    Oct 2008
    Posts
    156

    Cauchy Product

    Suppose  \sum a_{n}z^{n} and  \sum b_{n}z^{n} have radii of convergence  R_1 and  R_2 respectively. Show that the Cauchy Product  \sum c_{n}z^{n} converges for  |z| < \min(R_1, R_2) .

    So \sum c_nz^{n} = \sum_{k=0}^{n} a_{k}b_{n-k}z^{n}. Assume  |z| < \min(R_1, R_2) . Put  |z| = \min(R_1, R_2) - 2 \delta where  \delta < \min(R_1, R_2) . Then  \overline{\lim}|c_n|^{1/n} = \frac{1}{\min(R_1,R_2)} for  n large enough. Thus  |c_{n}z^{k}| \leq \left(\frac{\min(R_1,R_2)-2 \delta}{\min(R_1,R_2)- \delta} \right)^{k} . Thus  \sum  c_{n}z^{n} converges.

    Is this correct?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,541
    Thanks
    1394
    Quote Originally Posted by manjohn12 View Post
    Suppose  \sum a_{n}z^{n} and  \sum b_{n}z^{n} have radii of convergence  R_1 and  R_2 respectively. Show that the Cauchy Product  \sum c_{n}z^{n} converges for  |z| < \min(R_1, R_2) .

    So \sum c_nz^{n} = \sum_{k=0}^{n} a_{k}b_{n-k}z^{n}. Assume  |z| < \min(R_1, R_2) . Put  |z| = \min(R_1, R_2) - 2 \delta where  \delta < \min(R_1, R_2) . Then  \overline{\lim}|c_n|^{1/n} = \frac{1}{\min(R_1,R_2)} for  n large enough. Thus  |c_{n}z^{k}| \leq \left(\frac{\min(R_1,R_2)-2 \delta}{\min(R_1,R_2)- \delta} \right)^{k} . Thus  \sum  c_{n}z^{n} converges.

    Is this correct?

    Yes, it is. Notice, however, that this does NOT say that the radius of convergence is min(R1,R2). It is possible that the Cauchy product of two power series converges outside the radius of convergence of one of them.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: October 3rd 2011, 03:25 AM
  2. Cauchy Product
    Posted in the Calculus Forum
    Replies: 1
    Last Post: May 2nd 2010, 12:53 AM
  3. Cauchy Product
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: April 7th 2010, 04:27 AM
  4. An inverse question on Cauchy product of series
    Posted in the Differential Geometry Forum
    Replies: 6
    Last Post: August 27th 2009, 11:17 AM
  5. [SOLVED] What is the product of a cauchy and a normal distribution?
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: January 30th 2007, 06:11 AM

Search Tags


/mathhelpforum @mathhelpforum