I'm looking for the 3rd.-order Taylor approximation of
$(x,y,z) \mapsto e^x\cos(y)\sin(z)$ at $(x_0,y_0,z_0) = (0,0,0)$
I've got this piece of advice at hand: $\quad\textit{Use the Taylor series of known functions.}$
I can conclude that:
$\exp(x) = \sum_{n = 0}^\infty \frac{x^n}{n!}$
$\sin (x) = \sum_{n=0}^\infty (-1)^n\frac{x^{2n+1}}{(2n+1)!}$
$\cos (x) = \sum_{n=0}^\infty (-1)^n\frac{x^{2n}}{(2n)!}$
But how am I supposed to use this?
I get as a result by hiting it in Python (SymPy) :
$z + x*z + (x^2*z)/2 – (y^2*z)/2 – z^3/6$
Adding all three 2-dimensional taylor-approximations with n = 3 doesn't yield the same…
Any hints – not solutions at first- would be greatly appreciated.
Best Answer
To get what Python says try again by expanding $$\left(1 + x +x^2/2 + x^3/6\right)\left(1-y^2/2\right)\left(z - z^3/6\right)$$ and dropping monomials with degree more than $3$ (like $x^3z/6$ whose degree is $4$).