William Gosper
A few series
We have a general formula for x
less than 1:
where _{2}F_{1}
is a hypergeometric seie, which distract us from our topic, so I won't
mention it....
For x=1/2, we get :
which has a convergence of 2n
For we can write :
of convergence 3.39n
William Gosper is also used to formulae that are a bit weird making use
of , donc ask me of their use!
For example :
and by generalising :
Slice of life
William Gosper is part of those passionate about Pi
and of the small irreducible group that make computers suffer! Gosper
had in fact already calculated 17,001,303 terms of the continue
fraction of Pi in 1977. And then he programmed Ramanujan's serie (the one with 1103)
to obtain 17 millions decimals. We did not know then if this
serie converged, but the calculation was correct and only the number 1103 was not justified. So, as the Borwein said, the same as two different
integers whose difference is less than one must be equal, the number 1103
was found to satisfy. Because all other numbers leads on decimals
completly wrong dues to the sensitivity of errors...
Ah yes, still today, we have sometime to be contented with empirical
proofs!
Nevertheless let us not take away the scientifical force of William
Gosper, working on artificial intelligence, and on a small paquet of
properties concerning factorial
series converging to Pi
with R. Schroeppel. Unfortunatly I don't have the proof for the one
above, but it must be very similar to the proof below.
I don't if it is him and Schroeppel which found :
=
which seems to me a bit funny taking into account the
relative simplicity of the serie and the proof!
Around
First of all, notice that the factorial serie at the top of
the page will be perfecct for a spigot algorithm giving Pi, something that I will developpe
later on, on a new page.
But since we are talking of a factorial specialist, let's use this
opportunity to look at the other factorial series buzzing around Pi
!
Note that for more
simplicity...
So we get :
The formula (1) comes from the great Euler
while that (2) comes from Comtet in 1974.
The similarity between those two sum with the sum of power inverses is
quite stunning...
And of course, there exists no result for odd powers!
Proof
The proof for factorial series is a simple and same method,
typical of freshers, that I give briefly here for the case of the
formula :
=. By the way it's the same
procedure that I used to rediscover the proof of Katahiro's
formula...
If we consider .
This power serie has a convergence radius of 4. If we denote
hence the result is immediat by
Alembert's criteria.
Since it's a power serie we can derive without any problem, and se we
have:
The solving of the differential equation is just procedure, so I won't
write it's details.... We get the following general and particular
solutions:
We can deduce from the first result .
The we differentiate f and we get :
which ends the proof...
Hence, to get the general formula :
, we could notice that the members
of the right and left hand side of the first equality satisfy the
differential equation:
(1x^{2})f'(x)xf(x)=1 for
x<1
Trials
All those sequences have linear convergence or very close to
it (like a.n+b.Ln(n)) because the term in the serie decreases
in c^{n}.
Let's check this !

Gosper sequence

Sequence from the proof

Sequence (2) Comtet 
n=5 
3,14159249 (6) 
3,1306 (1) 
3,1415911 (5) 
n=10 
12 correct decimals

3,14157 (4) 
3,14159265340 (9) 
n=50 
57 correct decimals

28 correct decimals 
35 correct decimals

n=100 
? 
58 correct decimals

66 correct decimals

Pretty conclusive, no?
Gosper sequence has a convergence of roughly 1,2n, which is
quite good for this kind of sequence !
The sequence from the proof has 0,58n and the sequence of
Comtet (2) has 0,66n.
Acceleration of the convergence
As always with sequence with linear convergence, the Delta2
by Aitken proofs to be useful, but a bit less
than what one could hope for.

Suite de Gosper 
Suite de la démo 
Suite (2) Comtet 
n=5 
3,1415926507 (8) 
3,14195 (3) 
3,1415924 (6) 
n=10 
14 correct decimals

3,14159277 (6) 
11 correct decimals

n=50 
61 correct decimals

31 correct decimals

38 correct decimals

n=100 
? 
62 correct decimals 
69 correct decimals

34 extra decimals, which is not great, so we need to be
content with pushing the series one or two extra strps no more!
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