Table of Maclaurin Series Expansions

Below are the Maclaurin series expansions for several fundamental functions.

Exponential Function

$$ e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots + \frac{x^n}{n!} + o(x^n) $$

Natural Logarithm

$$ \log(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \dots + \frac{(-1)^{n+1} x^n}{n} + o(x^n) $$

Sine Function

$$ \sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots + \frac{(-1)^n x^{2n+1}}{(2n+1)!} + o(x^{2n+1}) $$

Cosine Function

$$ \cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots + \frac{(-1)^n x^{2n}}{(2n)!} + o(x^{2n}) $$

Tangent Function for |x| < π/2

$$ \tan(x) = x + \frac{x^3}{3} + \frac{2x^5}{15} + \dots + o(x^6) $$

Secant Function for |x| < π/2

$$ \sec(x) = 1 + \frac{x^2}{2} + \frac{5x^4}{4!} + \dots + o(x^n) $$

Arcsine Function for |x| < 1

$$ \arcsin(x) = x + \frac{x^3}{6} + \frac{3x^5}{40} + \frac{5x^7}{112} + \frac{35x^9}{1152} + o(x^9) $$

Arccosine Function for |x| < 1

$$ \arccos(x) = \frac{\pi}{2} - x + \frac{x^3}{6} - \frac{3x^5}{40} + \frac{5x^7}{112} - \frac{35x^9}{1152} + o(x^9) $$

Arctangent Function for |x| < 1

$$ \arctan(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \frac{x^9}{9} + o(x^9) $$