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
\[\begin{equation} \begin{aligned} (\sum_i a_ix^i)(\sum_j b_jx^j) &= \sum_i \sum_j a_ib_jx^{i+j} \\ &= \sum_{i,j, \; i+j=k} c_k x^k \end{aligned} \end{equation}\]




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

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