## Math-Linux.com

Knowledge base dedicated to Linux and applied mathematics.

Home > Mathematics > Taylor series expansion > Taylor series expansion of a function about 0

# Taylor series expansion of a function about 0

Below are the main Taylor series expansion around 0. These specific taylor developments are called Maclaurin development

## Landau notation, little o

Let $f$ be a function defined in the neighborhood of 0. For $\mathrm{n} \in \mathbb{N}^{*},$ $f$ grows much slower than $x^{n}$

$$f(x)=o\left(x^{n}\right) \Longleftrightarrow\forall \varepsilon>0, \exists \eta \in >0, \quad \forall x \in ]-\eta, \eta[ , \quad \left|f(x)\right| < \varepsilon \left|x^{n}\right|$$

## Maclaurin series - Taylor series expansion of a function about 0

\begin{aligned} e^{x} &=1+\frac{x}{1 !}+\frac{x^{2}}{2 !}+\cdots+\frac{x^{n}}{n !}+o\left(x^{n}\right) \\ \cos x &=1-\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}+\cdots+(-1)^{n} \cdot \frac{x^{2 n}}{(2 n) !}+o\left(x^{2 n+1}\right) \\ \sin x &=x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}+\cdots+(-1)^{n} \cdot \frac{x^{2 n+1}}{(2 n+1) !}+o\left(x^{2 n+2}\right) \\ \tan x &=x+\frac{x^{3}}{3}+\frac{2}{15} x^{5}+\frac{17}{315} x^{7}+o\left(x^{8}\right) \\ (1+x)^{\alpha} &=1+\alpha x+\frac{\alpha(\alpha-1)}{2 !} x^{2}+\cdots+\frac{\alpha(\alpha-1) \cdots(\alpha-n+1)}{n !} x^{n}+o\left(x^{n}\right) \\ \frac{1}{1-x} &=1+x+x^{2}+\cdots+x^{n}+o\left(x^{n}\right) \\ \frac{1}{1+x} &=1-x+x^{2}+\cdots+(-1)^{n} x^{n}+o\left(x^{n}\right) \\ \sqrt{1+x} &=1+\frac{x}{2}-\frac{1}{8} x^{2}+\cdots+(-1)^{n-1} \cdot \frac{1.1 \cdot 3 \cdot 5 \cdots(2 n-3)}{2^{n} n !} \\ \frac{1}{\sqrt{1+x}} &=1-\frac{x}{2}+\frac{3}{8} x^{2}+\cdots+(-1)^{n} \cdot \frac{1.3 \cdot 5 \ldots(2 n-1)}{2^{n} n !} x^{n}+o\left(x^{n}\right) \\ \ln (1+x) &=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}+\cdots+(-1)^{n-1} \cdot \frac{x^{n}}{n}+o\left(x^{n}\right) \\ \operatorname{ch} x &=1+\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}+\cdots+\frac{x^{2 n}}{(2 n) !}+o\left(x^{2 n+1}\right) \\ \operatorname{sh} x &=x+\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}+\cdots+\frac{x^{2 n+1}}{(2 n+1) !}+o\left(x^{2 n+2}\right) \\ \operatorname{th} x &=x-\frac{x^{3}}{3}+\frac{2}{15} x^{5}-\frac{17}{315} x^{7}+o\left(x^{8}\right) \\ \operatorname{argth} x &=x+\frac{x^{3}}{3}+\frac{x^{5}}{5}+\cdots+\frac{x^{2 n+1}}{2 n+1}+o\left(x^{2 n+2}\right) \\ \arctan x &=x-\frac{x^{3}}{3}+\frac{x^{5}}{5}+\cdots+(-1)^{n} \cdot \frac{x^{2 n+1}}{2 n+1}+o\left(x^{2 n+1}\right) \\ \operatorname{argsh} x &=x-\frac{1}{2} \frac{x^{3}}{3}+\frac{3}{8} \frac{x^{5}}{5}+\cdots+(-1)^{n} \cdot \frac{1.3 \cdot 5 \cdot \ldots(2 n-1)}{2^{n} n !} \frac{x^{2 n+1}}{2 n+1}+o\left(x^{2 n+2}\right) \\ \arcsin x &=x+\frac{1}{2} \frac{x^{3}}{3}+\frac{3}{8} \frac{x^{5}}{5}+\cdots+\frac{1.3 \cdot 5 \ldots(2 n-1)}{2^{n} n !} \frac{x^{2 n+1}}{2 n+1}+o\left(x^{2 n+2}\right) \end{aligned}