# Multiplication of power series

1. Apr 19, 2010

### sbashrawi

1. The problem statement, all variables and given/known data

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
2. Relevant equations

3. The attempt at a solution

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

2. Apr 19, 2010

### 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 $$d_n = a_n b_n$$?