1) Nilakantha Somayaji;
$\pi=3+\dfrac{4}{3^3-3}-\dfrac{4}{5^3-5}+\dfrac{4}{7^3-7}-\dfrac{4}{9^3-9}+.....$
2)Franciscus Vieta;
$\pi=2.\dfrac{2}{\sqrt2}.\dfrac{2}{\sqrt{2+\sqrt2}}.\dfrac{2}{\sqrt{2+\sqrt{2+\sqrt2}}}.\dfrac{2}{\sqrt{2+\sqrt{2+\sqrt{2+\sqrt2}}}}.....$
3)Gregory-Leibniz;
$\pi=\displaystyle\sum_{n=0}^{\infty}\dfrac{(-1)^n}{2n+1}$
4)Isaac Newton $\pi=\displaystyle\sum_{n=0}^{\infty}\dfrac{2^{(n+1)}.(n!)^2}{(2n+1)!}$
5)
6)
$\vdots$
This formulas ,from where come? How they did successfully finder formula of $\pi$
Can we create like these formulas?
I know ,my english not well,I hope I can explain myself.respects...
Precious Nilakantha's formula! In general from $f(x)$ if you have some value for which $af(x_0)=\pi$ you can developpe $f(x)$ in series and you get a formula $$\pi=a\sum a_nx_0^n$$ This general viewpoint could give some troubles, in particular the value of $x_0$ could be not good, the series cannot be convergent or not useful because convergence very slow (this occurs for example with Gregory-Leibniz's formula). The first in obtaining fast convergence was Machin putting the angle $\frac {\pi}{4}$ in function of smaller angles; he discovered in 1706 the formula $$\frac {\pi}{4}=4\arctan(\frac 15)-\arctan(\frac{1}{239})$$ in which with the values $x=\frac 15$ and $x=\frac{1}{239}$ give for the formula of Gregory an algorithm of fast convergence obtaining this way $100$ correct first digits for $\pi$.