# How to show$\frac{\pi}{4} = \frac{2\cdot4\cdot4\cdot6\cdot6\cdot8 \dotsm}{3\cdot3\cdot5\cdot5\cdot7\cdot7 \dotsm}$?

I am doing the exercises of Structure and Interpretation of Computer Programs. In exercise 1.31 the following equation is casually shown as an approximation of $\pi$:

$$\frac{\pi}{4} = \frac{2\cdot4\cdot4\cdot6\cdot6\cdot8 \dotsm}{3\cdot3\cdot5\cdot5\cdot7\cdot7 \dotsm}$$

How can this possibly be? Can someone provide me with more information on this approximation?

• Look up Wallis Product. – coffeemath Jan 9 '15 at 12:50
• See this link – Casteels Jan 9 '15 at 12:50
• Thanks! Remains a very strange fact that pi can be approximated like that. I am spacing out while trying to wrap my head around it. – Erwin Rooijakkers Jan 9 '15 at 13:04
• @user2609980 another cool one is $$\frac{\pi}{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}\ldots$$ – graydad Jan 9 '15 at 15:31
• @user2609980 Also, the brilliant mathematician Srinivasa Ramanujan was famous for finding things like $$\frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum^\infty_{k=0} \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}.$$ – graydad Jan 11 '15 at 18:09