The infinite product in this case is the same as the infinite sum of all integral square, namely the zeta function evaluated at 2.
Thus, (and I think you meant this)
Hence, pi appears in this expression.
But it's derived from measuring a quarter of the perimeter of an oddly shaped polygon inscribed in a unit circle
I must say that if you take away the equal sign noone (not even me) would guess that this thing is even remotely related to pi
after 100 terms, excel gives: 3.141299
After 200 however, excel gives: 3.194243
The way I set up the equation, it should never get above pi...
Can anyone verify excels answer (with mathematica or something)
If it's my equation, then I'll post the method for you guys to see what's wrong (I've gone over the work several times)
Consider a unit circle surrounding the origin, it's equation is
Now my method only uses the first quadrant, so forget all the other ones
First, put equally spaced points along the x-axis from 0 to 1.
Now draw vertical lines from the points.
Connect the intersections of the lines and the circle to each other (you'll get a quarter of an oddly shaped polygon)
So try finding the total of those measurements.
To do that you must first solve for the change in y (which I call for each increment.
Consider the vertical line at point , it intersects the circle at the y value:
Working through you'll eventually get:
Now the value of the hypotenuse of the triangle with as the height is:
And then solve...
Can you do it from here? It wasn't a very good explanation...
I dont get this ... as far as I can see, you calculate from the equation , to put an integral on , to get ... pi? How is that...
Isn't it simpler just integrating directly:
And one thing, I spoted that you set , that's equal to for real numbers .