Credit for creative kids, the picture above is the Halloween pumpkin carved by Sonja, one of my HS kids last year.

I know there are lots of infinite products that are equal to pi, and always thought

Wallis' expansion for pi/2 was beautiful,

And of course it is not as pretty, at least not to me, but since it was the first infinite product in math, I feel compelled to mention that Viete gave one for the reciprocal, 2/pi:

The question about this one for the really clever student is what does it have to do with the old calculus limit of Sin(theta)/theta... or the half angle formulas?

And Leibniz (you remember, the guy who invented Calculus if you DON'T live in England,

*just Kidding folks, it was*) wrote one that Clifford Pickover calls "eye candy for pi";

**Newton**all the way, good job Ikepi/4 = 1 - 1/3 + 1/5 - 1/7+ 1/9....

except, it takes forever to converge.... the sum of the first 250 terms is not accurate to the second decimal place...

Ok, but the point of all this........

But today I came across one I had never seen, from the master of us all, Euler. Euler, it seems wrote pi/2 as an infinite product of fractions in which the numerators were all prime and the denominators were all even numbers excluding multiples of four. What appears from what I see is that each denominator is one more or less than the prime in the numerator, but always avoiding the one which would be a multiple of four... (Ok, now how do you write that as in product notation??).

pi/2 = 3/2 ( 5/6)(7/6)(11/10) (13/14)(17/18)

I looked for this a little and could not come up with a reference. If anyone knows where Euler wrote this, please advise.

Franz Gnaedinger who wrote the post where I picked this up, also pointed out that

"The analogous infinite product using all odd numbers

in the numerator seems to approximate the natural

logarithm of 2:

ln2 = 1/2 x 3/2 x 5/6 x 7/6 x 9/10 x 11/10 ..."