University of Cambridge
S.C. Woon
A new algorithm
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= 
a_{n} 
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= 
a_{n} 
Delta2(a_{n}) 
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!
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