Let \(f(x)=\sum_{k=0}^{\infty} a_k(x-x_0)^k\) and \(g(x)=\sum_{k=0}^{\infty} b_k(x-x_0)^k\) be two power series . Then, on their common domain we have

\begin{equation*} f(x)g(x)=\sum_{k=0}^{\infty} \sum_{i+j=k} a_i b_j (x-x_0)^k. \end{equation*}

[1, Proposition 1.1.7]

The same is true for multidimensional power series [1, Proposition 2.2.2].

Remarks

References Link to heading

  1. S. Krantz and H. Parks, A Primer of Real Analytic Functions. Boston, MA: Birkhäuser Boston, 2002. doi:10.1007/978-0-8176-8134-0