• Arithmetics

Arithmetics

$$a^n-b^n=(a-b)(a^{n-1}+a^{n-2}b+\cdots +a{b}^{n-2}+b^{n-1})$$

$$a^3+b^3=(a+b)(a^2-ab+b^2)$$

$$\begin{gathered} \log xy=\log x+\log y \\ \log x^r=r\log x \end{gathered}$$

$$\frac{\sqrt{a^2+b^2}-b}{a^2}=\frac{a^2}{\sqrt{a^2+b^2}+b}$$

sigma notation

$$\sum _{i=m}^{n}f(x_i)$$

• linear

common sum

$$\begin{gathered} \sum _{i=1}^{n}i=\frac{n(n+1)}{2} \\ \sum _{i=1}^n{i}^2=\frac{n(n+1)(2n+1)}{6} \\ \sum _{i=1}^n{i}^3={\left[\frac{n(n+1)}{2}\right]}^2 \end{gathered}$$

trigonometric transformation

$$\begin{gathered} \sin A\cos B=\frac{1}{2}[\sin (A-B)+\sin (A+B)] \\ \sin A\sin B=\frac{1}{2}[\cos (A-B)-\cos (A+B)] \\ \cos A\cos B=\frac{1}{2}[\cos (A-B)+\cos (A+B)] \end{gathered}$$

$\lfloor x\rfloor$ greatest integer smaller or equal than $x$

$$\underset{n\to \infty }{\lim }\frac{A^{\prime }}{B^{\prime }}=1\Rightarrow \underset{n\to \infty }{\lim }{\left(1+\frac{r}{B}\right)}^A=e^r$$

binomial coefficient

$$\left(\begin{array}{c}k\\ n\end{array}\right)=\frac{k(k-1)\cdots (k-n+1)}{n!}$$

triangle inequality

$$|a+b|\le |a|+|b|$$

reverse triangle inequality

$$||x|-|y||\le |x-y|$$

power when $|z|<<1$

$$(1+z{)}^n=1+nz+\cdots$$