www.pi314.net Boris Gourévitch The world of Pi - V2.57 modif. 13/04/2013
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Gottfried Wilhelm Leibniz
(1646 - 1716)

Why always do complicated?

(simply !)

and a derived sequnce:

Parts of his life:

Leibniz was born in Leipzig in 1646. Although German mathematician and philosopher, all his work was written in French or Latin. Son of a philosophy professor, he received very little mathematical teaching, but he discovered a hidden interest for mathematics thanks to Huygens with who he forms a friendship... (Thanks Huygens!).

Leibniz was especially interested in numerical series - That is interesting ! - and claims to be the father of the differential calculus, which will lead him to a violent dispute with Newton and darken the remaining of his life.

We do not always know this, but this good old Leibniz was a great notation inventor. we owe him the (integral) sign, = , dx , . for the multiplication and division.

But Leibniz, although he was a genius, did not have a neat idea about complex numbers for example, and wrote lines and lines of equation without any real meaning, like:

with which he astonished Huygens.

His philosophic conceptions influenced a lot his mathematical vision of things... He considered for example these complex numbers as between existence and non existence!

Ignored at his death by the German and British scientific communities, the famous French hundred-year-old Fontenelle actually praised him in Paris... He was worth it!

Around

Here I am exaggerating a bit, because Leibniz was not the real discoverer of this formula. James Gregory (1638-1675) had in fact calculated the whole sequence expansion:

arctan(x)=

for x between-1 and 1. It would be astonishing if Grégory had not seen the particular case x=1 which gives the following formula:

But here, as we will see, it's convergence is more than execrable and Grégory must have realized how useless this formula is practically... But admit that it is beautiful, and extremely simple! So let us give Grégory what he deserves, even if the first publication of this formula is only explicitly shown in Leibniz's works...
On the other hand, this formula from Leibniz/Grégory reserves a few surprises, discovered only recently by Roy North... Go and see about them in applications !

Demonstration

We can remind ourselves that we have arctan:: R->]-/2,/2 is defined as the inverse function of the bijection tan : ]-/2,/2[->R.

But we have tan'(x)=1+tan2(x) (using cos and sin ) and hence, from the formula of the derivative of an inverse function.
But we also know the whole sequence defined on]-1,1[. We know that we can integrate it term by term on this interval from the properties of whole sequences. After seeing that the result converges at 1 and -1 which insures the uniform convergence of the sequence over[-1,1] towards arctan, we take x=1 and that's it!... We can also use x=1/31/2, and that is more interesting:

Applications

Behold, this is a great moment!
For the Leibniz/Grégory formula:

 n=10 3,2323 (0) n=100 3,151493401 (1) n=1 000 3,14259165 (2) n=10 000 3,1416926435905432 (3) n=100 000 3,14160265348979398846014336 (3) n=500 000 3,141590653589793240462643383269502884197 (5) n=1 000 000 3,14159365358879323921264313 (5)

If there are a few digits in brown, it is because they are incorrect! But then, how come the following digits be right? Is the convergence not logarithmic from its form (approx. log(n)) ? That is the problem that Roy North asked the Borwein brothers a few years ago... I have not got the solution, but it involves Euler 's numbers and two summing formulae. Still waiting for further answers...

For the second formula (just over the Application section), we have an interesting linear convergence of about n/2 :

 n=10 3,1415933045 n=100 49 digits correct

Acceleration of the convergence :

If there is a formula for which Aitken's Delta2 is really really useful, then it is Leibniz's... And other acceleration methods work as well! (a logarithmic convergence is not very convincing)...

1) With the Delta2 : classical acceleration, and faster and faster normally... requires a computation with a lot of digits because of the unstable numerical side of the process.

2) With an average:
As it is a modifies sequence, I had the idea of using of applying an average value to Leibniz's series. What should of brought us in theory 1 digit in the best case seems to accelerate the convergence in a quite surprising way!:
let be the la moyenne pondérée associée.

The results appear in the table below for the 2 types of acceleration:

 Leibniz Delta2 Average value n=3 2,8952 (0) 3,13333 (1) 3,161904 (1) n=5 2,976 (0) 3,13968 (1) 3,143434 (2) n=10 3,2323 (0) 3,141839 (3) 3,14150053 (4) n=100 3,15149 (1) 3,141592905 (6) 3,14159264593 (7) n=1000 3,142591 (2) 3,141592653839792 (9) 3,141592653589041 (12) n=10000 3,14169264 (3) 3,1415926535900 ? (10) 3,1415926535897931634 (15)

The last result for Delta2 seems suspicious to me, according to the average value found, although I calculated it with 100 digits...

One good thing about series made up from others, is that we can repeat the process.
Immediate application!
 Delta2 repeated 2 times Average repeated 2 times n=3 3,13888 (1) does not exist... n=5 3,141450 (2) 3,1215 (1) n=10 3,141595655 (5) 3,141598653 (5) n=100 3,1415926536094 (9) 3,141592653589922 (12) n=1000 3,1415926535897934269 (15) 3,14159265358979323847405 (19) n=10000 too suspicious! 3,14159265358979323846264338440 (26)

Just note that combinations between Delta2 and means are less efficient.
3) we can also rearrange the terms as Leibniz had done, but it is not much more efficient.

All right, this time I think that's all concerning this sequence that proved in the end to be quite rich!!!