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| Gotfried Wilhelm von Leibniz |
Two posts ago I
mentioned that the best simple fractional approximation of the value of p is
355/113 = 3.14159292..., which was discovered in the West in the 16th century. Later,
better approximations were obtained, but no
longer in the form of a fraction, rather as the sum of a series.
Several infinite series of terms are known whose sum is p. So it
is enough to add a sufficiently large number of terms to obtain as many digits
of p as we
want, as long as we have time to do the sums. The first to propose one of these
series was the French mathematician François Vieta. As his series was quite complicated,
we give here the much better known series proposed in 1673 by the German mathematician
and philosopher Gotfried Wilhelm von Leibniz:
The more terms we add of this series, the
closer we will come to the value of p. The following table shows the
advances made over time in the calculation of the successive approximations of
this number, using different series, formulas or procedures.
