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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 $5$-degree Taylor polynomial for function $f(x)= \arctan \left(x\right)$ in $x_0=0$.

#### Function and value of the function

$f(x)= \arctan \left(x\right)$,      $f(0)= 0$

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

$i$$f^{(i)}(x)$$f^{(i)}(x_0)$$\frac{f^{(i)}(x_0)}{i!} 1 {{1}\over{x^2+1}}$$ 1 $$1 2 -{{2{}x}\over{\left(x^2+1\right)^2}}$$ 0 $$0 3 {{2{}\left(3{}x^2- 1\right)}\over{\left(x^2+1\right)^3}}$$ -2 $$-{{1}\over{3}} 4 -{{24 {}\left(x-1\right){}x{}\left(x+1\right)}\over{\left(x^2+1\right)^4}}$$ 0 $$0 5 {{24{}\left(5{}x^4-10{}x^2+1\right)}\over{\left(x^2+1 \right)^5}}$$ 24$${{1}\over{5}}$

#### Polinomios de Taylor

$T_{5}(x)= x-{{1}\over{3}}{}x^3+{{1}\over{5 }}{}x^5$