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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

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