Taylor/Mclaurin Series of a Product
\[f(x), \; g(x)\]
about a point by collecting powers of \[(x-x_0)\]
or \[x\]
in the product of the Taylor or Mclaurin series.If the Taylor series for
\[f(x)\]
is \[\sum^{\infty}_{n=0} a_n(x-x_0)^n\]
and the Taylor series for \[g(x)\]
us \[\sum^{\infty}_{m=0} b_m(x-x_0)^m\]
then the Taylor series for \[f(x)g(x)\]
is \[\sum^{\infty}_{k=0} c_k (x-x_0)^k c\]
where \[c_k= \sum_{m+n=k} a_nb_m\]
and similarly for the Mclaurin series.Example: Find the Mclaurin series for
\[e^x sinx\]
up to \[x^3\]
.\[e^x=\sum^{\infty}_{n=0} \frac{x^n}{n!}=1 + x+ \frac{x^2}{2!}+ \frac{x^3}{3!}+...=1+x+ \frac{x^2}{2}+ \frac{x^3}{6}+...\]
\[sinx=\sum^{\infty}_{m=0} \frac{(-1)^mx^{2m+1}}{(2m+1)!}=x - \frac{x^3}{3!}+...=x- \frac{x^3}{6}+...\]
\[\begin{equation} \begin{aligned} e^xsinx &= (1+x+ \frac{x^2}{2}+ \frac{x^3}{6}+...)(x- \frac{x^3}{6}+...) \\ &= x+(1)x^2+( \frac{-1}{6} + \frac{1}{2} )x^3+... \\ &= x+x^2 + \frac{x^3}{3}+.... \end{aligned} \end{equation} \]
