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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 $