constants can seem a bit magical when we do not know how to compute
them. This page and others are about computing pi to so many places, so
we know how to obtain it. There are many ways of computing pi.
Scotsman James Gregory (1638-1675) was an exceptionally talented
mathematician who is credited with the discovery of the arctangent
series, called the Gregory Series (sometimes called the Leibniz
series). Apparently, he knew the Binomial Theorem for fractions (before
Newton published it), and had the basics of the calculus, and
conceivably could have been the one to develop it. He knew Taylor's
series 40 years before Taylor published it, and he also knew the series
for tan x, sec x, arctan x, and arcsec x, although he is remembered
only for the arctan series.
He knew the area under the curve y=1/(1+x2)
between 0 and x was arctan x:
So, a particular example is:
to Leibniz principle that an alternating series, with the absolute
value of each term less than its predecessor, and its terms tending to
zero, is a convergent series. The maximum error in the sum of such a
series is equal to the value of the last omitted term. The Gregory
series is such a series.
needed for 4 decimal places
The series will be accurate to 4 decimal places when the last omitted
term is less than 0.00005
Actually, rounding errors bring the needed number of terms up to
80,000-100,000 for 4 decimal places of accuracy.
the series is impractical for the calculation of pi, we can accelerate
it. Considering the days when arithmetic had to be done by hand, it
seems that the Gregory Series might be a better series to use under
Taking the sum of k terms of an
alternating series is less accurate than taking the sums to two
consecutive terms and taking the average.
Repeating the process produces less error.
The table below shows the results of calculating the sum to 12...21
terms, and progressively taking the average of these sums.
The table above shows that the arithmetic can be kept
relatively simple when using the Gregory series, by successively taking
the averages of the sums. Using 21 terms and successively taking the
average, we find pi to approximately 9 decimals (compare with 80000
terms required for 4 decimal places!)
Continuing the above scheme to 30 terms resulted in a pi
estimation of 3.14159265358979, which was as far as the spreadsheet