Taylor and Maclaurin Series

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Taylor and Maclaurin Series

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Taylor Series for Common Functions

We already know the power series for $\displaystyle\frac{1}{1+x}$ , $\ln(1+x)$ , $\displaystyle\frac{1}{(1-x)^2}$ , and $\tan^{-1}(x)$ . Using the representation formula in Taylor's Theorem,

$f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} (x-a)^n,$ we can derive the Taylor series expansion for a number of other common functions:

$\displaystyle e^x = \sum_{n=0}^\infty \frac{x^n}{n!} = 1+x+\frac{x^2}{2!} + \frac{x^3}{3!} + \ldots$

$\displaystyle e^{2x} = \sum_{n=1}^\infty \frac{2^n}{n!} x^n$

$\displaystyle e^{x^2} = \sum_{n=0}^\infty \frac{x^{2n}}{n!}$

$\displaystyle\sin(x) = \sum_{n=0}^\infty \frac{(-1)^{n} x^{2n+1}}{(2n+1)!} =x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \ldots$

$\displaystyle\cos(x) = \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!} x^{2n} = 1 - \frac{x^2}{2} + \frac{x^4}{4!} - \frac{x^6}{6!} + \ldots$

$\displaystyle\sqrt{1+x} = 1 + \frac{x}{2} - \frac{x^2}{8} + \frac{x^3}{16} + \ldots$