François Viete
(1540 - 1603)

A first infinite product concerning Pi

that is to say :

Slice of life

François Viete, born in Fontenay-le-compte in 1540 firstly achieved his law diploma in 1960. But concurrently of his juridical career, he is interested in sciences. In that way he wrote an astonomy treatise ("Principes de cosmographie"). But  as he was banished in 1584 for 5 years, having some "free time" he starts to take interest in mathematics !
It was his most productive period. Working on algebra and mainly on polynomials, he rather simplifies notations and discovers a third-degree-equation solving method. He is also interested in the expression of cos(nx) and sin(nx) in terms of cos(x) and sin(x), he obtained what we call today Tchebychev's polynomials.

About

His researches in trigonometry induces him to compute nine decimals of Pi. But particularly, Viete states the first infinite known product equivalent to Pi ! Unfortunately, he does not worry about the effective convergence of his product. This one will be prooved in the nineteenth century. Viete does not use the perimeters as Archimedes but the polygons' area.

Proof

Both algorithms in the top of the page are equivalent, that is immediate by recurrence. After that there are two methods. The lazy's which consist to notice that both algorithmes are equivalents to Archimedes' series where will have been multiplied by its conjugate (+ instead of the first minus) then we simplify. And there is the geometrical construction; weighty but interesting for being closest of the mathematician's original inspiration, which I present here:

Viete is interested in regular polygons' area inscribed in a circle of unity radius of which the sides would have been doubled each time (that is amazing!).

This area would finish to tend toward circle's, that is to say .
To do that, he divides a n sides polygon whose he compute the area.

                                  We have : OH=Rcos() and IJ=2JH=2Rsin()
Well...
Then we note An the n sides polygon's area:

If the number of the polygon's sides is doubled, we divide by 2 and this new area is equal to:


Let put :

A the circle's area is given by A=R²= so =A_n/P.
but if we make k tend to infinite, P is written as a cosine product.
(because in this case, A_kn -> A) so we can write :

Now, we know that :


A_n being the first polygon's area with n=4 (square =p/4), we have A_n=n.sin()cos()=2 et U₀=2^-1/2
therefore we get :

Notice that if we start from another polygon, we obtain an other expression (as profs would say in theirs syllabi, I leave that for the attention of the reader !)

Product convergence :

This result is quite simple but essential :



(obviously, that would not have sense normally to note the sum if it does not exist, but the study of probability show me that there was not any problem to write that, particularly in that branch of learning since the sum can equal infinity, without any fuss... And I do not mind to write that if it is understandable) the product converges equally so the series have a sense.

Trials

Hum, if the Viete's infinite product is obtained thanks to Archimedes', normally we will be confronted with a linear convergence:

n=5	3,141277 (3)
n=10	3,1415923 (6)
n=50	30 decimals true
n=100	60 decimals true

No problem ! A great convergence in 3n/5 as for Archimedes.

Convergence speeding up

And as for this old good Archimedes, there is a very efficient convergence speedind up by Aitken's Delta2 :

n=5	3,14159280 (6)
n=10	3,14159265358993 (12)
n=50	60 decimals true

Ok, nothing less than a doubling of the convergence speed !!!