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66.4: Useful Series

Last updated

Aug 10, 2024
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- 66.3: Hyperbolic Trigonometry
- 66.5: Table of Derivatives

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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 $x$ .

$\begin{aligned} & (1+x)^{1 / 2}=1+\dfrac{1}{2} x-\dfrac{1}{8} x^{2}+\dfrac{1}{16} x^{3}-\dfrac{5}{128} x^{4}+\dfrac{7}{256} x^{5}-\dfrac{21}{1024} x^{6}+\dfrac{33}{2048} x^{7}-\dfrac{429}{32768} x^{8}+\cdots \text { (D.1) } \\[10pt] & (1-x)^{1 / 2}=1-\dfrac{1}{2} x-\dfrac{1}{8} x^{2}-\dfrac{1}{16} x^{3}-\dfrac{5}{128} x^{4}-\dfrac{7}{256} x^{5}-\dfrac{21}{1024} x^{6}-\dfrac{33}{2048} x^{7}-\dfrac{429}{32768} x^{8}-\cdots \text { (D.2) } \\[10pt] & (1+x)^{-1 / 2}=1-\dfrac{1}{2} x+\dfrac{3}{8} x^{2}-\dfrac{5}{16} x^{3}+\dfrac{35}{128} x^{4}-\dfrac{63}{256} x^{5}+\dfrac{231}{1024} x^{6}-\dfrac{429}{2048} x^{7}+\dfrac{6435}{32768} x^{8}-\cdots \text { (D.3) } \\[10pt] & (1-x)^{-1 / 2}=1+\dfrac{1}{2} x+\dfrac{3}{8} x^{2}+\dfrac{5}{16} x^{3}+\dfrac{35}{128} x^{4}+\dfrac{63}{256} x^{5}+\dfrac{231}{1024} x^{6}+\dfrac{429}{2048} x^{7}+\dfrac{6435}{32768} x^{8}+\cdots \text { (D.4) } \\[10pt] & \dfrac{1}{a+x}=\dfrac{1}{a}-\dfrac{x}{a^{2}}+\dfrac{x^{2}}{a^{3}}-\dfrac{x^{3}}{a^{4}}+\dfrac{x^{4}}{a^{5}}-\dfrac{x^{5}}{a^{6}}+\cdots \\[10pt] & e^{x}=1+x+\dfrac{x^{2}}{2}+\dfrac{x^{3}}{6}+\dfrac{x^{4}}{24}+\dfrac{x^{5}}{120}+\dfrac{x^{6}}{720}+\dfrac{x^{7}}{5040}+\dfrac{x^{8}}{40320}+\dfrac{x^{9}}{362880}+\cdots \\[10pt] & \sin x=x-\dfrac{x^{3}}{6}+\dfrac{x^{5}}{120}-\dfrac{x^{7}}{5040}+\dfrac{x^{9}}{362880}-\dfrac{x^{11}}{39916800}+\dfrac{x^{13}}{6227020800}-\cdots \\[10pt] & \cos x=1-\dfrac{x^{2}}{2}+\dfrac{x^{4}}{24}-\dfrac{x^{6}}{720}+\dfrac{x^{8}}{40320}-\dfrac{x^{10}}{3628800}+\dfrac{x^{12}}{479001600}-\cdots \end{aligned}$

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