Question

ordem quatro do polinómio de Taylor f(x) = ln(x)

Answer 1

$f(x)=\ln x\\ \\ f'(x)=\dfrac{1}{x}\\ \\ f''(x)=-\dfrac{1}{x^2}\\ \\ f'''(x)=\dfrac{2}{x^3}\\ \\ f^{iv}(x)=-\dfrac{6}{x^4}$

$\displaystyle f(x)\sim \sum_{k=0}^n\dfrac{f^{k)}(x_0)}{k!}(x-x_0)^k+o(x-x_0)^k\\ \\ f(x)\sim \ln x_0+\dfrac{(x-x_0)}{x_0}-\dfrac{(x-x_0)^2}{2x_0^2}+\dfrac{(x-x_0)^3}{3x_0^3}-\dfrac{(x-x_0)^4}{4x_0^4}+\cdots+\\ \\ \hspace*{9.2cm}o(x-x_0)^4$

donde $x_0\ \textgreater \ 0$