Formulae sheet series

The formulae booklet contains the following Maclaurin series of standard functions.

Exponential function

\(e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots + \frac{x^n}{n!} + \cdots, \quad x \in \mathbb{R}. \)

Sinusoidal functions

\( \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots + \frac{(-1)^k}{(2k+1)!} x^{2k+1} + \cdots, \quad x \in \mathbb{R}.\)

\(\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots + \frac{(-1)^k}{(2k)!} x^{2k} + \cdots, \quad x \in \mathbb{R}.\)

Translated natural logarithm

\(\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} + \cdots + \frac{(-1)^{k+1}}{k} x^k + \cdots, \quad -1 < x \leq 1.\)

Inverse tangent formula

\(\arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} + \cdots + \frac{(-1)^{k+1}}{2k-1} x^{2k-1} + \cdots, \quad -1 < x < 1. \)