Reportar fallo

## 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

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