Let

\[ \begin{aligned} f(x) &= \sum_{n=0}^\infty a_n x^n \\ g(x) &= \sum_{n=0}^\infty b_n x^n \\ R &= \text{the minimum of their radii of convergence} \\ \end{aligned} \]

Then

\[ \begin{aligned} f(x)\,g(x) &= \sum_{n=0}^\infty (\sum_{i=0}^{n}a_i\, b_{n-i}) \, x^n \\ \end{aligned} \]

for \(\displaystyle x \in (-R, R)\) !

You can find more information at Calculus Two: Sequences and Series.

# posted by rot13(Unafba Pune) @ 1:53 PM