Celebrating infinitely the day of Pi

A few Pi decimal ones

Since you will know many of you today, on March 14, it is the day of Pi. if someone does not know why, the reason is that in the Anglo-Saxon world the dates he writes himself of the form Month / day / year. Thus today would be 3/14.

Every year I write something related to Pi this day. And this year is not going to be less. We are going to celebrate the day of Pi of infinite form.

Of infinite form?

We are going to celebrate this day of Pi of infinite form showing diverse sums and infinite products where this wonderful number appears. We go with them:

• As it seems, it was François Viète who gave the first exact numerical expression in which Pi appears. Specifically it was this infinite product:

  

• This expression, also like infinite product, was discovered by John Wallis:

• The famous sum of the problem of Basel (and II) discovered by Leonhard Euler:

• But not much less this was supreme the only expression related to Pi discovered by Euler. Big Leonhard found also expressions of the previous type at least: up to exponent 26!!. For exponent 4 we have this expression:

  And for exponent 6 this one:

• Pero Euler discovered many other infinite expressions, so much supreme like products, related to Pi. Some of them are the following ones:

  In her the numerators of the fractions are the prime numbers except 3 and the denominators take a sum when the prime number is of the form and a subtraction when it performs the form.

  Here the odd numbers appear like denominators and the signs are alternated + and - between the fractions.

  And in this expression they appear in the denominators of the squares of all the odd numbers that are not multiple of 3.

• Newton discovered the following expression related to Pi:

• From certain results discovered by Euler we can come to the following relation:

• Further on in the time, specifically in 1997, Bailey found the following sum on Pi:

  

• Separate chapter is deserved by the expressions related to Pi discovered by Ramanujan. For example:

  

  I recommend the linkage to MathWorld that appears at the end of the article to see other expressions of this style which discoverer was Ramanujan.

• And to finish I leave to you a monster of numerical expression discovered by the brothers Chudnosky. It is one of the most powerful expressions at the time of calculating decimal of Pi (he calculates 14 decimal ones exactly in every step).

  It is the following one:

I have left to myself many expressions which protagonist is Pi. If you know someone that should not appear in this article and believe that it is important or interesting do not hesitate to write it in the comments.

Other days of Pi in Gaussianos:

• The day of Pi and The day of Pi II in 2007.
• How to demonstrate that Pi is irrational (II) in 2008.
• Celebrating the day of Pi with a needle and a jellyfish in 2009.

Sources:

• History of the mathematician, of Carl B. Boyer.
• Introductio in Analysin Infinitorum, of Leonhard Euler.
• Pi you formulate in MathWorld.
• The image that illustrates this article is extracted of this Flickr set.