\[\sum_i a_ix^i, \; \sum_j b_jx^j\]
which converge on intervals \[I_1, \; I_2\]
respectively. The Cauchy product of the two power series is\[\begin{equation} \begin{aligned} (\sum_i a_ix^i)(\sum_j b_jx^j) &= \sum_i \sum_j a_ib_jx^{i+j} \\ &= \sum_{i,j, \; i+j=k} c_k x^k \end{aligned} \end{equation}\]
.Then
\[c_0=a_0b_0\]
\[c_1=a_0b_1+a_1b_0\]