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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 $