He knew the area under the curve y=1/(1+x

[1.1]

Giving:

[1.2]

So, a particular example is:

[1.3]

According 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.

[1.4]

Actually, rounding errors bring the needed number of terms up to 80,000-100,000 for 4 decimal places of accuracy.

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.

Term | Value | Average 1 | Average 2 | Average 3 | Average 4 | Average 5 | Average 6 | Average 7 | Average 8 | Average 9 |

12 | 3.0584027659273300 | |||||||||

13 | 3.2184027659273300 | 3.1384027659273300 | ||||||||

14 | 3.0702546177791900 | 3.1443286918532600 | 3.1413657288902900 | |||||||

15 | 3.2081856522619400 | 3.1392201350205600 | 3.1417744134369100 | 3.1415700711636000 | ||||||

16 | 3.0791533941974300 | 3.1436695232296800 | 3.1414448291251200 | 3.1416096212810200 | 3.1415898462223100 | |||||

17 | 3.2003655154095500 | 3.1397594548034900 | 3.1417144890165900 | 3.1415796590708600 | 3.1415946401759400 | 3.1415922431991200 | ||||

18 | 3.0860798011238300 | 3.1432226582666900 | 3.1414910565350900 | 3.1416027727758400 | 3.1415912159233500 | 3.1415929280496400 | 3.1415925856243800 | |||

19 | 3.194187909231940 | 3.140133855177880 | 3.141678256722290 | 3.141584656628690 | 3.141593714702260 | 3.141592465312810 | 3.141592696681220 | 3.141592641152800 | ||

20 | 3.091623806667840 | 3.142905857949890 | 3.141519856563890 | 3.141599056643090 | 3.141591856635890 | 3.141592785669080 | 3.141592625490940 | 3.141592661086080 | 3.141592651119440 | |

21 | 3.189184782277600 | 3.140404294472720 | 3.141655076211300 | 3.141587466387600 | 3.141593261515340 | 3.141592559075610 | 3.141592672372350 | 3.141592648931640 | 3.141592655008860 | 3.141592653064150 |

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 would go.

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