Continued Fractions

I was working on some continued fractions, and came across the fundamental recurrence formula above. It worked pretty well for some simple continued fractions (like √2) but seemed to be a bit unfriendly towards calculating π.

I'm using this definition of π:


Now, using that definition with the above equation, it was always completely wrong...

So I tried changing the equation...

So you'll notice a few specific changes... 1) A₁ has been redefined in terms of previous results. 2) A₁ reduced the index on a₁ to a₀. 3) A_n+1 changed the index on a_n+1 to a_n. 4) B_n+1 changed the index on b_n+1 to b_n.

This may seem like a lot of changes, but really #1 didn't change the outcome at all and the rest were all about reducing the index of a/b.

Net result? All the previous stuff (√2, golden mean, etc) give the same results as before... but now π works.