Suppose you take 1/A, and subtract 1/B from it. Well, it gets smaller, right? But there's another way to make 1/A smaller: increase A. Specifically, replace A by A + x. If you choose x shrewdly, you can make 1/(A + x) = 1/A - 1/B. What x does the trick?
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And now it's clear there's no stopping us. We next replace 1/C by 1/C - 1/D = 1/(C + z), and repeat. And so on, and so forth, ad infinitum. We get a continued fraction from an alternating series of reciprocals. This is how Euler derived Brouncker's continued fraction for 4/pi.
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