hello all
i have a bit problem with finding the taylor polynomial for a composed function.
if anyone can help me find the taylor polynomial for the function f(x) = e^sin(x) and how i would appriciate it alot
thanks!
It's in general not easy to find the Taylor expansion of the composition of two function. There exists a formula, due to Faà di Bruno.
You can always do it the long way. Let $\displaystyle \displaystyle \begin{align*} e^{\sin{x}} = c_0 + c_1x + c_2x^2 + c_3x^3 + \dots \end{align*}$.
We know that $\displaystyle \displaystyle \begin{align*} e^{\sin{0}} = 1 \end{align*}$, so let $\displaystyle \displaystyle \begin{align*} x = 0 \end{align*}$ and we find $\displaystyle \displaystyle \begin{align*} c_0 = 1 \end{align*}$.
Differentiate both sides and we find
$\displaystyle \displaystyle \begin{align*} e^{\sin{x}}\cos{x} &= c_1 + 2c_2x + 3c_3x^2 + 4c_4x^3 + \dots \end{align*}$
We know $\displaystyle \displaystyle \begin{align*} e^{\sin{0}}\cos{0} = 1 \end{align*}$, so let $\displaystyle \displaystyle \begin{align*} x = 0 \end{align*}$ and we find $\displaystyle \displaystyle \begin{align*} c_1 = 1 \end{align*}$.
Differentiate both sides and we have
$\displaystyle \displaystyle \begin{align*} e^{\sin{x}}\cos^2{x} - e^{\sin{x}}\sin{x} = 2c_2 + 2\cdot 3c_3x + 3\cdot 4c_4x^2 + 4\cdot 5c_5x^3 + \dots \end{align*}$
We know that $\displaystyle \displaystyle \begin{align*} e^{\sin{0}}\cos^2{0} - e^{\sin{0}}\sin{0} = 1 \end{align*}$ so let $\displaystyle \displaystyle \begin{align*} x = 0 \end{align*}$ and we find $\displaystyle \displaystyle \begin{align*} c_2 = \frac{1}{2} \end{align*}$.
Follow this process to find as many terms as you need. I doubt that there will be a pattern.