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Polinomios de Taylor

$ n$-degree Taylor polynomial for the function $ f(x)$ around $ x=x_0$\[\begin{align*} T_n(x)&=f(x_0)+f'(x_0)(x-x_0)+\frac{f''(x_0)}{2!}(x-x_0)^2+\frac{f'''(x_0)}{3!}(x-x_0)^3+\cdots\\&+\frac{f^{(i)}(x_0)}{i!}(x-x_0)^i+\cdots \frac{f^{(n)}(x_0)}{n!}(x-x_0)^n \end{align*}\]

Problema

Find $ 4 $-degree Taylor polynomial for function $ f(x)= \ln \left(\cos \left(x\right)\right) $ in $ x_0=0$.

Function and value of the function

$ f(x)= \ln \left(\cos \left(x\right)\right) $,      $ f(0)= 0 $

Derivatives, derivatives at $0$ and coefficients of Taylor polynomial

$ i$$f^{(i)}(x)$$f^{(i)}(x_0)$$\frac{f^{(i)}(x_0)}{i!}$
1$ -{{\sin \left(x\right)}\over{\cos \left(x\right)}} $$ 0 $$ 0 $
2$ -{{1}\over{ \cos ^2\left(x\right)}} $$ -1 $$ -{{1}\over{2}} $
3$ -{{2{}\sin \left(x \right)}\over{\cos ^3\left(x\right)}} $$ 0 $$ 0 $
4$ {{4}\over{\cos ^2 \left(x\right)}}-{{6}\over{\cos ^4\left(x\right)}} $$ -2 $$ -{{1 }\over{12}} $

Polinomios de Taylor

$ T_{4}(x)= -{{1}\over{2}}{}x^2-{{1}\over{12}}{}x^4 $