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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 $ 9 $-degree Taylor polynomial for function $ f(x)= {{1}\over{1+x^2}} $ in $ x_0=0$.

Function and value of the function

$ f(x)= {{1}\over{1+x^2}} $,      $ f(0)= 1 $

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

$ i$$f^{(i)}(x)$$f^{(i)}(x_0)$$\frac{f^{(i)}(x_0)}{i!}$
1$ -{{2{}x}\over{\left(x^2+1\right) ^2}} $$ 0 $$ 0 $
2$ {{2{}\left(3{}x^2-1\right)}\over{\left(x^2+1\right)^3 }} $$ -2 $$ -1 $
3$ -{{24{}\left(x-1\right){}x{}\left(x+1\right)}\over{ \left(x^2+1\right)^4}} $$ 0 $$ 0 $
4$ {{24{}\left(5{}x^4-10{}x^2+1\right) }\over{\left(x^2+1\right)^5}} $$ 24 $$ 1 $
5$ -{{240{}x{}\left(x^2-3 \right){}\left(3{}x^2-1\right)}\over{\left(x^2+1\right)^6}} $$ 0 $$ 0 $
6$ {{720{}\left(7{}x^6-35{}x^4+21{}x^2-1\right)}\over{\left(x^2+1 \right)^7}} $$ -720 $$ -1 $
7$ -{{40320{}\left(x-1\right){}x{}\left(x+1 \right){}\left(x^2-2{}x-1\right){}\left(x^2+2{}x-1\right)}\over{ \left(x^2+1\right)^8}} $$ 0 $$ 0 $
8$ {{40320{}\left(3{}x^2-1\right){} \left(3{}x^6-27{}x^4+33{}x^2-1\right)}\over{\left(x^2+1\right)^9}} $$ 40320 $$ 1 $
9$ -{{725760{}x{}\left(x^4-10{}x^2+5\right){}\left(5{}x^ 4-10{}x^2+1\right)}\over{\left(x^2+1\right)^{10}}} $$ 0 $$ 0 $

Polinomios de Taylor

$ T_{9}(x)= 1-x^2+x ^4-x^6+x^8 $