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Taylor and Maclaurin Series

Flashcards

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Function: $\sqrt{1+x}$ at $x=0$

$1 + \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16}x^3 - \frac{5}{128}x^4 + \cdots$

Function: $\cosh(x)$ at $x=0$

$1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \frac{x^6}{6!} + \cdots$

Function: $\cos(x)$ at $x=0$

$1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots$

Function: $\tan^{-1}(x)$ at $x=0$

$x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots$

Function: $\sin(xy)$ at $(x, y)=(0, 0)$

$xy - \frac{(xy)^3}{3!} + \frac{(xy)^5}{5!} - \frac{(xy)^7}{7!} + \cdots$

Function: $\sin(x)$ at $x=0$

$x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$

Function: $e^{ax}$ at $x=0$ , where $a$ is a constant

$1 + ax + \frac{a^2x^2}{2!} + \frac{a^3x^3}{3!} + \frac{a^4x^4}{4!} + \cdots$

Function: $\frac{1}{1+x^2}$ at $x=0$

$1 - x^2 + x^4 - x^6 + x^8 - \cdots$

Function: $e^x$ at $x=0$

$1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots$

Function: $\cos(x)$ at $x=\frac{\pi}{4}$

$\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}(x - \frac{\pi}{4}) - \frac{\sqrt{2}}{4}(x - \frac{\pi}{4})^2 + \frac{\sqrt{2}}{6}(x - \frac{\pi}{4})^3 + \cdots$

Function: $\sinh(x)$ at $x=0$

$x + \frac{x^3}{3!} + \frac{x^5}{5!} + \frac{x^7}{7!} + \cdots$

Function: $\ln(x+1)$ about $x=1$

$\ln(2) + \frac{1}{2}(x-1) - \frac{1}{4}(x-1)^2 + \frac{1}{6}(x-1)^3 - \cdots$

Function: $(1+x)^n$ at $x=0$ , n is any real number

$1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \cdots$

Function: $\ln(1+x)$ at $x=0$

$x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots$

Function: $\frac{1}{1-x}$ at $x=0$

$1 + x + x^2 + x^3 + x^4 + \cdots$

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