66.4: Useful Series
- Page ID
- 91934
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The first four series are valid if \(|x|<1\); the fifth is valid for \(x^{2}<a^{2}\); and the last three are valid for all real \(x\).
\[
\begin{aligned}
& (1+x)^{1 / 2}=1+\dfrac{1}{2} x-\dfrac{1}{8} x^{2}+\dfrac{1}{16} x^{3}-\dfrac{5}{128} x^{4}+\dfrac{7}{256} x^{5}-\dfrac{21}{1024} x^{6}+\dfrac{33}{2048} x^{7}-\dfrac{429}{32768} x^{8}+\cdots \text { (D.1) } \\[10pt]
& (1-x)^{1 / 2}=1-\dfrac{1}{2} x-\dfrac{1}{8} x^{2}-\dfrac{1}{16} x^{3}-\dfrac{5}{128} x^{4}-\dfrac{7}{256} x^{5}-\dfrac{21}{1024} x^{6}-\dfrac{33}{2048} x^{7}-\dfrac{429}{32768} x^{8}-\cdots \text { (D.2) } \\[10pt]
& (1+x)^{-1 / 2}=1-\dfrac{1}{2} x+\dfrac{3}{8} x^{2}-\dfrac{5}{16} x^{3}+\dfrac{35}{128} x^{4}-\dfrac{63}{256} x^{5}+\dfrac{231}{1024} x^{6}-\dfrac{429}{2048} x^{7}+\dfrac{6435}{32768} x^{8}-\cdots \text { (D.3) } \\[10pt]
& (1-x)^{-1 / 2}=1+\dfrac{1}{2} x+\dfrac{3}{8} x^{2}+\dfrac{5}{16} x^{3}+\dfrac{35}{128} x^{4}+\dfrac{63}{256} x^{5}+\dfrac{231}{1024} x^{6}+\dfrac{429}{2048} x^{7}+\dfrac{6435}{32768} x^{8}+\cdots \text { (D.4) } \\[10pt]
& \dfrac{1}{a+x}=\dfrac{1}{a}-\dfrac{x}{a^{2}}+\dfrac{x^{2}}{a^{3}}-\dfrac{x^{3}}{a^{4}}+\dfrac{x^{4}}{a^{5}}-\dfrac{x^{5}}{a^{6}}+\cdots \\[10pt]
& e^{x}=1+x+\dfrac{x^{2}}{2}+\dfrac{x^{3}}{6}+\dfrac{x^{4}}{24}+\dfrac{x^{5}}{120}+\dfrac{x^{6}}{720}+\dfrac{x^{7}}{5040}+\dfrac{x^{8}}{40320}+\dfrac{x^{9}}{362880}+\cdots \\[10pt] & \sin x=x-\dfrac{x^{3}}{6}+\dfrac{x^{5}}{120}-\dfrac{x^{7}}{5040}+\dfrac{x^{9}}{362880}-\dfrac{x^{11}}{39916800}+\dfrac{x^{13}}{6227020800}-\cdots \\[10pt] & \cos x=1-\dfrac{x^{2}}{2}+\dfrac{x^{4}}{24}-\dfrac{x^{6}}{720}+\dfrac{x^{8}}{40320}-\dfrac{x^{10}}{3628800}+\dfrac{x^{12}}{479001600}-\cdots
\end{aligned}
\]