Sunday, 17 May 2009
Euler's infinite Expansion of Pi with primes
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 Newton all the way, good job Ike) wrote one that Clifford Pickover calls "eye candy for pi";
pi/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 ..."