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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 $f(x)= \sqrt{1+x}$ in $x_0=0$.

Function and value of the function

$f(x)= \sqrt{1+x}$,      $f(0)= 1$

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

$i$$f^{(i)}(x)$$f^{(i)}(x_0)$$\frac{f^{(i)}(x_0)}{i!} 1 {{{{1}\over{2}}}\over{\sqrt{x+1}}}$$ {{ 1}\over{2}} $${{1}\over{2}} 2 -{{{{1}\over{4}}}\over{\left(x+1 \right)^{{{3}\over{2}}}}}$$ -{{1}\over{4}} $$-{{1}\over{8}} 3 {{{{3 }\over{8}}}\over{\left(x+1\right)^{{{5}\over{2}}}}}$$ {{3}\over{8}} $${{1}\over{16}} 4 -{{{{15}\over{16}}}\over{\left(x+1\right)^{{{7 }\over{2}}}}}$$ -{{15}\over{16}}$$-{{5}\over{128}}$

Polinomios de Taylor

$T_{4}(x)= 1+{{1}\over{2 }}{}x-{{1}\over{8}}{}x^2+{{1}\over{16}}{}x^3-{{5}\over{128}}{}x^4$