Ejercicio Analisis Matematico 1 calculo MacLaurin

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Hola Ariel!

Desarrollo Mc Laurin:

$$\begin{align}&P(x)=f(0)+f'(0)+ \frac{f''(0)}{2!}x^2+\frac{f'''(0)}{3!}x^3+····+\frac{f^{(n)}(0)}{n!}+E_n\\&\\&derivadas:\\&f(x)=ln(1+x^2) ==> f(0)=0\\&\\&f'(x)= \frac{2x}{1+x^2}==>f'(0)=0\\&\\&f''(x)= \frac{2-2x^2}{(1+x^2)}==> f''(0)=2\\&\\&f'''(x)= \frac{4x^3-12x^2}{(1+x^2)^3}==> f'''(0)=0\\&\\&f^{(4)}(x)=\frac{-12x^4+72x^2-12}{(x^2+1)^4}==> f^{(4)}(0)=-12\\&\\&f^{(5)}(x)= \frac{48x^5-480x^3+240x}{(1+x^2)^6}==>f^{(5)}(0)=0\\&\\&f^{(6)}(x)= \frac{ -240x^6+3600x^4-3600x^2+240}{(1+x^2)^6}==> f^{(6)}(0)=240\\&\\&\\&P(x)=f(0)+f'(0)+ \frac{f''(0)}{2!}x^2+\frac{f'''(0)}{3!}x^3+····=\\&\\&0+0+ \frac 2 {2!}x^2+0- \frac {12}{4!} x^4+0+ \frac{240}{6!}x^6+·····=\\&\\&x^2- \frac 1 2 x^4+ \frac 1 3x^6+·········=  \sum _{n=1}^{ +\infty}\frac{x^{2n}} n· (-1)^{n+1}\\&\\&===>\\&a_n=\frac{(-1)^{n+1}} n\\&\\&\\&Criterio \de \Cauchy\\&\\&\lim_{n \to \infty} \frac{|a_{n+1}|}{|a_n|}=\lim_{n \to \infty} \frac{|\frac{(-1)^{n +2}}{n+1}|}{|\frac{(-1)^{n+1}}n|}=\lim_{n \to \infty} \frac{n}{n+1}=1=A\\&\\&Radio \convergencia= \frac 1 A=1\\&\\&intervalo \ convergencia:(-1,1)\\&\\&b)\\&ln( \frac 5 4)=ln(1+(\frac 1 2)^2)\\&\\&P( \frac 1 2)=(\frac 1 2)^2- \frac 1 2 (\frac 1 2)^4+ \frac 1 3 (\frac 1 2)^6+ E_7= \frac {43}{192}+E_7\\&\\&Error absoluto= \bigg| ln( \frac 5 4 )- \frac {43}{192} \Bigg|=8.147820191·10^{-4}\\&\\&Errorº Relativo=\frac {8.147820191·10^{-4}}{ln \frac 5 4}·100=0.3651 \  \%\\&\end{align}$$

Saludos

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