\[f(g(x))\]
of functions \[f(x), \; g(x)\]
about a point by collecting powers of \[(x-x_0)\]
or \[x\]
by using the Taylor or Mclaurin series of one function inside the Taylor or Mclaurin series of the other.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 \[f(g(x))\]
is \[\sum^{\infty}_{k=0} \frac{d^kf}{dx^k}(g(x)-g(x_0))^k\]
.Example: Find the Mclaurin series for
\[{e^{cosx}}\]
up to \[x^4\]
.\[e^x=\sum^{\infty}_{n=0} \frac{x^n}{n!}=1 + x+ \frac{x^2}{2!}+ \frac{x^3}{3!}+...=1+x+ \frac{x^2}{2}+ \frac{x^3}{6}+...\]
\[cpsx=\sum^{\infty}_{m=0} \frac{(-1)^{m}x^{2m}}{(2m)!}=1- \frac{x^2}{2!}+ \frac{x^4}{4!}-...\]
\[\begin{equation} \begin{aligned}e^{cosx} &= 1+((1- \frac{x^2}{2!}+ \frac{x^4}{4!}-...)-1)+ \frac{(1- \frac{x^2}{2!}+ \frac{x^4}{4!}-...)-1)^2}{2!}+... \\ &= 1- \frac{x^2}{2}+ \frac{5x^4}{24}+... \end{aligned} \end{equation}\]