Saltar al contenido principal
LibreTexts Español

A.5: Tabla de Expansiones de Taylor

  • Page ID
    118876
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

    ( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\id}{\mathrm{id}}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\kernel}{\mathrm{null}\,}\)

    \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\)

    \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\)

    \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    \( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

    \( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

    \( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vectorC}[1]{\textbf{#1}} \)

    \( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

    \( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

    \( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    Dejar\(n\ge \) ser un entero. Entonces si la función\(f\) tiene\(n+1\) derivadas en un intervalo que contiene ambos\(x_0\) y\(x\text{,}\) tenemos la expansión Taylor

    \[\begin{align*} f(x)&=f(x_0)+f'(x_0)\,(x-x_0)+\dfrac{1}{2!}f''(x_0)\,(x-x_0)^2+\cdots +\dfrac{1}{n!}f^{(n)}(x_0)\,(x-x_0)^n\\ &\hskip0.5in +\dfrac{1}{(n+1)!}f^{(n+1)}(c)\,(x-x_0)^{n+1}\qquad\hbox{for some $c$ between $x_0$ and $x$} \end{align*}\]

    El límite como\(n\rightarrow\infty\) da la serie Taylor

    \[\begin{align*} f(x)&=\sum_{n=0}^\infty\dfrac{f^{(n)}(x_0)}{n!}(x-x_0)^n \end{align*}\]

    para\(f\text{.}\) Cuando\(x_0=0\) esto también se llama la serie Maclaurin para\(f\text{.}\) Aquí están las expansiones de la serie Taylor de algunas funciones importantes.

    \[\begin{align*} e^x&=\sum_{n=0}^\infty \dfrac{1}{n!}x^n &&\text{for } -\infty \lt x \lt \infty\\ &=1+x+\dfrac{1}{2}x^2+\dfrac{1}{3!}x^3+\cdots+\dfrac{1}{n!}x^n+\cdots\\ \sin x&=\sum_{n=0}^\infty\dfrac{(-1)^n}{(2n+1)!}x^{2n+1} &&\text{for } -\infty \lt x \lt \infty\\ &=x-\dfrac{1}{3!}x^3+\dfrac{1}{5!}x^5-\cdots +\dfrac{(-1)^n}{(2n+1)!}x^{2n+1}+\cdots\\ \cos x&=\sum_{n=0}^\infty\dfrac{(-1)^n}{(2n)!}x^{2n} &&\text{for } -\infty \lt x \lt \infty\\ &=1-\dfrac{1}{2!}x^2+\dfrac{1}{4!}x^4-\cdots +\dfrac{(-1)^n}{(2n)!}x^{2n}+\cdots\\ \dfrac{1}{1-x}&=\sum_{n=0}^\infty x^n &&\text{for } -1\le x \lt 1\\ &=1+x+x^2+x^3+\cdots+x^n+\cdots\\ \dfrac{1}{1+x}&=\sum_{n=0}^\infty(-1)^n x^n &&\text{for } -1 \lt x\le 1\\ &=1-x+x^2-x^3+\cdots+(-1)^nx^n+\cdots\\ \ln(1-x)&=-\sum_{n=1}^\infty\dfrac{1}{n}x^n &&\text{for } -1\le x \lt 1\\ &=-x-\frac{1}{2} x^2-\dfrac{1}{3}x^3-\cdots-\dfrac{1}{n}x^n-\cdots\\ \ln(1+x)&=-\sum_{n=1}^\infty\dfrac{(-1)^n}{n}x^n &&\text{for } -1 \lt x\le 1\\ &=x-\frac{1}{2} x^2+\dfrac{1}{3}x^3-\cdots-\dfrac{(-1)^n}{n}x^n-\cdots\\ (1+x)^p&=1+px+\dfrac{p(p-1)}{2}x^2+\dfrac{p(p-1)(p-2)}{3!}x^3+\cdots\\ &\hskip0.5in+ \dfrac{p(p-1)(p-2)\cdots(p-n+1)}{n!}x^n+\cdots \end{align*}\]


    This page titled A.5: Tabla de Expansiones de Taylor is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Joel Feldman, Andrew Rechnitzer and Elyse Yeager via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.