University of Cambridge

S.C. Woon

Slice of life

S. C. Woon is a student researcher in the applied maths department of the University of Cambridge. And... that all I know about him! He has no personal webpage and his publication are still rare. The formula that interest us appeared in the presogious magazine American Mathematical Monthly in 1995 under the very explicit name of Problem 1441 !

Proof

So you want a proof? Here we go...
Show by recurrence that :

For n=0, no problem...
Suppose now that the result holds for a certain non zero natural number n.
We have by the first step of the proof by induction.

But,

hence by summing, the cot terms cancel each other out, cot(/2)=0 and we are left with

But we also have so we finaly get

and it is what we wanted...

So we conclude that by proof by induction which show us that the result is valid for all natural n.
Then, it is obvious that by equivalence of sin(x) and x in 0 that :

Trials

Is this weird formula useful by the way?
Well it's not too bad:

n=	an
5	3,14033 (2)
10	3,1415914215 (5)
20	11 correct decimals
50	29 correct decimals
100	59 correct decimals

Just a bit less than 0,6n but it's still a linear convergence, which is interesting!
The justification of the linear convergence can be taken from the proof on the page about Cues where the sequence b_n, correspond practicly to this page.

Acceleration of convergence

Like all good sequences with linear convergence, the Delta2 of Aitken accelerate efficiently our sequence. While it is the same result as on the page about Cues, I put them there so to avoid you changing page!

n=	an	Delta2(an)
5	3,14033 (2)	3,141595089 (5)
10	3,1415914215 (5)	3,1415926535921 (10)
20	11 correct decimals	23 correct decimals
50	29 correct decimals	59 correct decimals
100	59 correct decimals	...

I can still no go past the 100 decimals, for the moment I'm using more Mathematica instead of Student Maple that I use normaly...
Still the rapidity of convergence is doubled (nearly 1,2n), which is really impressive!