https://www.overleaf.com/learn/latex/Integrals%2C_sums_and_limits
https://hackmd.io/@CynthiaChuang/Basic-LaTeX-Commands

$a_0 x^0+a_1 x^1+a_2 x^2+...$
$\sum_{n=0}^{\infty} a_n x^n$
$\sum_{n=0}^{\infty} a_n x^n = a_0 x^0+a_1 x^1+a_2 x^2+...$

$\sum_{n=0}^{\infty} a_n x^n =A(x)$
$\sum_{n=0}^{\infty} a(n) x^n$
$\sum_{n=0}^{\infty} a_n x^n = \sum_{n=0}^{\infty} a(n) x^n =A(x)$

$a(n) \rightarrow A(x)$
$1 \rightarrow \frac{1}{1-x}$
$x^0+x^1+x^2+...$

$\frac{1}{n!} \rightarrow e^x$
$\frac{1}{0!}x^0+\frac{1}{1!}x^1+\frac{1}{2!}x^2+...$

$\sum_{n=0}^{\infty} a(n) x^n =A(x)$
$n = 0, 1 ,2...$
$\int_{0}^{\infty} a(n) x^n \,dn =A(x)$
$0{\le}t<{\infty}$
$\int_{0}^{\infty} a(t) x^t \,dt =A(x)$

$x^t = (e^{ln(x)})^t =e^{tln(x)} = e^{-st}$
$0<x<1$
$ln(x)<0$
$s = -ln(x)$
$s >0$
$\int_{0}^{\infty} a(t) e^{-st} \,dt =A(x)= A(e^{-s})$
$\int_{0}^{\infty} f(t) e^{-st} \,dt =F(s)$

$f(t) \rightarrow F(s)$