APPROXIMATE CALCULATION OF PI

It's known, that pi is transcendental and can't be accurately expressed using finite formulas or be calculated in any other way. In former times, some enthusiasts mathematicians calculated this value manually by the recurrence formulas, that sometimes resulted in an sad mistakes and nullified all of their longstanding efforts.

Now certainly, all of its approximate values are obtained by long and complex calculations with the use of computer technology. It has enabled to calculate the "pi" with an accuracy of many thousands of characters. Perhaps these characters are used somewhere in reality, but in general, such calculations are "gymnastics of the mind" for developers of algorithms, as well as allow you to use a sequence of digits of "pi" as a random number generator.

Below are the most simple formulas that allow to calculate, with varying degrees of approximation, the value of this amazing and somewhat magic number. Of course, the list of these formulas is infinite, as infinite the sequence of digits of "pi". However, it is possible that somebody can to prove finiteness of the number of such formulas? Worth to try...

      22 / 7       = 3.142857142857143...
     201 / 64      = 3.140625 
     333 / 106     = 3.141509433962264...
     355 / 113     = 3.141592920353983...
   75948 / 24175   = 3.141592554291623...
  100798 / 32085   = 3.141592644537946...
  103993 / 33102   = 3.141592653011902...
  312689 / 99532   = 3.141592653618936...
  833719 / 265381  = 3.141592653581078...
 4272943 / 1360120 = 3.141592653589389...
 5419351 / 1725033 = 3.141592653589815...
        PI         = 3.141592653589793.....