James Gregory’s sequence for π

shouldn’t be so quick. It converges very slowly and so doesn’t present an environment friendly strategy to compute π. After summing half 1,000,000 phrases, we solely get 5 right decimal locations. We will confirm this with the next `bc`

code.

s = 0 scale = 50 for(okay = 1; okay <= 500000; okay++) { s += (-1)^(k-1)/(2*k-1) } 4*s

which returns

3.14159__0__…

which differs from π within the sixth decimal place. So does that imply there’s nothing fascinating about Gregory’s system? Not so quick!

When anybody speaks of quite a lot of right decimals, they practically all the time imply variety of consecutive right digits following the decimal level. However **for this put up solely** I’ll take the time period **actually** to imply the variety of decimals that match the decimals within the right reply.

The variety of right decimals (on this non-standard use of the time period) within the sequence above shouldn’t be so unhealthy. Right here’s the consequence, with the digits that differ from these of π underlined:

3.14159__0__6535897932__40__4626433832__6__9502884197__2__…

So despite the fact that the sixth decimal worth is incorrect, the following 10 after which are right, after which after a pair errors we get one other string of right digits.

In [1] the authors clarify what makes this instance tick, and present the way to create comparable sequences. For instance, we see an analogous sample each time the restrict of the sum is half of an influence of 10, however not a lot for different limits. For instance, let’s enhance 500,000 to 600,000. We get

3.14159__0986923126572953384124016__

which is totally incorrect after the sixth digit. So despite the fact that the result’s barely extra correct, it has fewer right decimals.

## Associated posts

[1] Jonathan Borwein, Peter Borwein, Karl Dilcher. Pi, Euler Numbers, and Asymptotic Expansions. American Mathematical Month-to-month. vol 96, p. 681–687.