Multiplication of power series

[sbashrawi]

Homework Statement

Suppose that the power series \Sigma[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

Homework Equations

The Attempt at a Solution

I found that the convergence radius is > min(R1, R2)

 

[ystael]

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

Perhaps the question intended to refer to a different notion of product, such as \sum d_n z^n where d_n = a_n b_n?