## The Cauchy Product

Take two power series
$\sum_i a_ix^i, \; \sum_j b_jx^j$
which converge on intervals
$I_1, \; I_2$
respectively. The Cauchy product of the two power series is
\begin{aligned} (\sum_i a_ix^i)(\sum_j b_jx^j) &= \sum_i \sum_j a_ib_jx^{i+j} \\ &= \sum_{i,j, \; i+j=k} c_k x^k \end{aligned}
.
Then
$c_0=a_0b_0$

$c_1=a_0b_1+a_1b_0$

$c_2=a_0b_2+a_1b_1+a_2b_0$

$c_3=a_0b_3+a_1b_2+a_2b_1+a_3b_0$

The product of the two series will only converge if both series converge, i.e. on the intersection of
$I_1, \; I_2$
.