*third in a finite series on infinity
(see posts 1 and 2)*

Somehow, I suspect I wouldn’t survive long on the frontier.

Drop me in the American West, circa 1850, and I fear my math-blogging and bad-drawing skills might not carry me far. I need indoor plumbing. I need the rule of law. I need chain coffee shops. I’m not cut out for the frontier.

And yet the frontier is exactly where I found myself the other day, when I came across this formula in the wonderful *Penguin Book of Curious and Interesting Numbers*, by David Wells:

I decided to play around with this product a bit. After all, what are products for, if not playing around?

(Go ahead and play with your Apple products. I’ll play with my infinite ones. We’ll see who has more fun.)

I felt like there should be an easier way to write this expression, exploiting the repetition of factors, so I gave it a shot, and created this:

Then my brain exploded and the universe dissolved around me, because I had just punched logic in the face, and it had punched me back.

The left side of that equation is π/2. It’s roughly 1.57.

The right side of that equation, however, is a product of many numbers—*all of them below 1.*

What happens when you multiply two numbers smaller than 1? You get *another* number smaller than 1.

How the heck could that equal 1.57?