Taylor/Mclaurin Series of a Function of a Function

We can find the Taylor or Mclaurin series of a function of a function  
\[f(g(x))\]
  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  
\[x^4\]
.
\[e^x=\sum^{\infty}_{n=0} \frac{x^n}{n!}=1 + x+ \frac{x^2}{2!}+ \frac{x^3}{3!}+...=1+x+ \frac{x^2}{2}+ \frac{x^3}{6}+...\]

\[cpsx=\sum^{\infty}_{m=0} \frac{(-1)^{m}x^{2m}}{(2m)!}=1- \frac{x^2}{2!}+ \frac{x^4}{4!}-...\]

\[\begin{equation} \begin{aligned}e^{cosx} &= 1+((1- \frac{x^2}{2!}+ \frac{x^4}{4!}-...)-1)+ \frac{(1- \frac{x^2}{2!}+ \frac{x^4}{4!}-...)-1)^2}{2!}+... \\ &= 1- \frac{x^2}{2}+ \frac{5x^4}{24}+... \end{aligned} \end{equation}\]

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