Editing Power series (section)

=== Multiplication and division ===
With the same definitions for <math>f(x)</math> and <math>g(x)</math>, the power series of the product and quotient of the functions can be obtained as follows:
<math display="block">\begin{align}
  f(x)g(x) &= \biggl(\sum_{n=0}^\infty a_n (x-c)^n\biggr)\biggl(\sum_{n=0}^\infty b_n (x - c)^n\biggr) \\
           &= \sum_{i=0}^\infty \sum_{j=0}^\infty  a_i b_j (x - c)^{i+j} \\
           &= \sum_{n=0}^\infty \biggl(\sum_{i=0}^n a_i b_{n-i}\biggr) (x - c)^n.
\end{align}</math>

The sequence <math display="inline">m_n = \sum_{i=0}^n a_i b_{n-i}</math> is known as the [[Cauchy product]] of the sequences <math>a_n</math> and {{nowrap|<math>b_n</math>.}}

For division, if one defines the sequence <math>d_n</math> by
<math display="block">\frac{f(x)}{g(x)} = \frac{\sum_{n=0}^\infty a_n (x - c)^n}{\sum_{n=0}^\infty b_n (x - c)^n} = \sum_{n=0}^\infty d_n (x - c)^n</math>
then 
<math display="block">f(x) = \biggl(\sum_{n=0}^\infty b_n (x - c)^n\biggr)\biggl(\sum_{n=0}^\infty d_n (x - c)^n\biggr)</math>
and one can solve recursively for the terms <math>d_n</math> by comparing coefficients.

Solving the corresponding equations yields the formulae based on [[determinant]]s of certain matrices of the coefficients of <math>f(x)</math> and <math>g(x)</math>
<math display="block">d_0=\frac{a_0}{b_0}</math>
<math display="block">d_n=\frac{1}{b_0^{n+1}} \begin{vmatrix}
a_n    &b_1   &b_2   &\cdots&b_n    \\
a_{n-1}&b_0   &b_1   &\cdots&b_{n-1}\\
a_{n-2}&0     &b_0   &\cdots&b_{n-2}\\
\vdots &\vdots&\vdots&\ddots&\vdots \\
a_0    &0     &0     &\cdots&b_0\end{vmatrix}</math>