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Series Expansions

Notation: $ k!= k\,(k-1)\,(k-2)..2.1$ , $ f^{(n)}(x)=d^nf(x)/dx^n$ .

$\displaystyle f(x)$ $\displaystyle = f(a) + \frac{(x-a)}{1!}\,f^{(1)}(a) + \frac{(x-a)^2}{2!}\,f^{(2)}(a)+\cdots$ (1226)
$\displaystyle (1+x)^\alpha$ $\displaystyle = 1 + \alpha x + \frac{\alpha (\alpha-1)}{2!} x^2 + \frac{\alpha (\alpha-1) (\alpha-2)}{3!}  x^3+\cdots$ (1227)
$\displaystyle {\rm e}^x$ $\displaystyle = 1 + x + \frac{x^2}{2!}+ \frac{x^3}{3!} + \cdots$ (1228)
$\displaystyle \ln(1+x)$ $\displaystyle = x - \frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots$ (1229)
$\displaystyle \sin x$ $\displaystyle =x - \frac{x^3}{3!} +\frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$ (1230)
$\displaystyle \cos x$ $\displaystyle = 1-\frac{x^2}{2!} + \frac{x^4}{4!}-\frac{x^6}{6!} + \cdots$ (1231)
$\displaystyle \tan x$ $\displaystyle = x + \frac{x^3}{3} + \frac{2 x^5}{15} + \frac{17 x^7}{315}+\cdots$ (1232)
$\displaystyle \sinh x$ $\displaystyle = x + \frac{x^3}{3!} +\frac{x^5}{5!} + \frac{x^7}{7!} + \cdots$ (1233)
$\displaystyle \cosh x$ $\displaystyle = 1+\frac{x^2}{2!} + \frac{x^4}{4!}+\frac{x^6}{6!} + \cdots$ (1234)
$\displaystyle \tanh x$ $\displaystyle = x - \frac{x^3}{3} + \frac{2 x^5}{15} - \frac{17 x^7}{315}+\cdots$ (1235)

     
     



Richard Fitzpatrick 2013-04-08