How to Find the Taylor Polynomial of a Function Composition?

[IMDerek]

Is there any nice trick for finding the Taylor polynomial of a composition of 2 functions, both of which can be expressed as taylor polynomials themselves? For example, finding the taylor polynomial for $$e^{\cos x}$$. Thanks.

 

[AiRAVATA]

Well, for example, near [itex]\pi/2[/tex]<br /> <br /> $$e^{\cos x}=1+\cos x+ \frac{cos^2 x}{2!}+\frac{\cos^3 x}{3!}+...$$<br /> <br /> and<br /> <br /> $$\cos x=-\frac{(x-\pi/2)^2}{2!}+\frac{(x-\pi/2)^4}{4!}-...$$<br /> <br /> now, the hard part is to compose it, so maybe it's easier to just calculate the derivative and evaluate, depends on what are you looking for.[/itex]

 

[Gib Z]

There is in fact a nice trick which cleverly works out the coefficients (and hence the whole series): http://en.wikipedia.org/wiki/Taylor_series#Calculation_of_Taylor_series