A.5: Tabla de Expansiones de Taylor
- Page ID
- 119214
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.
\[\begin{align*} e^x&=\sum_{n=0}^\infty \dfrac{1}{n!}x^n &&\text{for } -\infty \lt x \lt \infty\\ &=1+x+\dfrac{1}{2}x^2+\dfrac{1}{3!}x^3+\cdots+\dfrac{1}{n!}x^n+\cdots\\ \sin x&=\sum_{n=0}^\infty\dfrac{(-1)^n}{(2n+1)!}x^{2n+1} &&\text{for } -\infty \lt x \lt \infty\\ &=x-\dfrac{1}{3!}x^3+\dfrac{1}{5!}x^5-\cdots +\dfrac{(-1)^n}{(2n+1)!}x^{2n+1}+\cdots\\ \cos x&=\sum_{n=0}^\infty\dfrac{(-1)^n}{(2n)!}x^{2n} &&\text{for } -\infty \lt x \lt \infty\\ &=1-\dfrac{1}{2!}x^2+\dfrac{1}{4!}x^4-\cdots +\dfrac{(-1)^n}{(2n)!}x^{2n}+\cdots\\ \dfrac{1}{1-x}&=\sum_{n=0}^\infty x^n &&\text{for } -1\le x \lt 1\\ &=1+x+x^2+x^3+\cdots+x^n+\cdots\\ \dfrac{1}{1+x}&=\sum_{n=0}^\infty(-1)^n x^n &&\text{for } -1 \lt x\le 1\\ &=1-x+x^2-x^3+\cdots+(-1)^nx^n+\cdots\\ \ln(1-x)&=-\sum_{n=1}^\infty\dfrac{1}{n}x^n &&\text{for } -1\le x \lt 1\\ &=-x-\frac{1}{2} x^2-\dfrac{1}{3}x^3-\cdots-\dfrac{1}{n}x^n-\cdots\\ \ln(1+x)&=-\sum_{n=1}^\infty\dfrac{(-1)^n}{n}x^n &&\text{for } -1 \lt x\le 1\\ &=x-\frac{1}{2} x^2+\dfrac{1}{3}x^3-\cdots-\dfrac{(-1)^n}{n}x^n-\cdots\\ (1+x)^p&=1+px+\dfrac{p(p-1)}{2}x^2+\dfrac{p(p-1)(p-2)}{3!}x^3+\cdots\\ &\hskip0.5in+ \dfrac{p(p-1)(p-2)\cdots(p-n+1)}{n!}x^n+\cdots \end{align*}\]