Tag Archives: continued fractions

Astounding “Natural” Identities

The modest sequence \(1,2,3,4,\ldots\) can do some rather awe-inspiring things, when properly arranged. Here’s a short list of some of its many impressive feats.

There are numerous expressions for \(\pi\) relying on the progression of integers, including the Wallis formula: $$\frac{\pi}{2} = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdots$$ (which can be derived from Complex Analysis using an infinite product representation for the sine function) and an elegant alternating sum: $$\frac{\pi-3}{4} = \frac{1}{2\cdot 3\cdot 4}-\frac{1}{4\cdot 5\cdot 6}+\frac{1}{6\cdot 7\cdot 8}-\cdots$$ (try to prove this!). Euler’s number \(e\) has similarly surprising formulas, such as $$\frac{1}{e-2} = 1+\frac{1}{2+\frac{2}{3+\frac{3}{4+\frac{4}{5+\cdots}}}}$$ (listed at MathWorld) and $$\frac{e}{e-1} = 1+\frac{1+\frac{1+\frac{1+\cdots}{2+\cdots}}{2+\frac{2+\cdots}{3+\cdots}}}{2+\frac{2+\frac{2+\cdots}{3+\cdots}}{3+\frac{3+\cdots}{4+\cdots}}}$$ (which is problem 1745 in Mathematics Magazine, posed by Gerald A. Edgar).

The list doesn’t stop here! The nested square-root identity $$3 = \sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots}}}}$$ is attributed to Ramanujan (on this Wikipedia page). As another curiosity, the sequence $$\frac{1}{2},\qquad
\frac{1}{2} \Big/ \frac{3}{4},\qquad
\frac{\frac{1}{2} \big/ \frac{3}{4}}{\frac{5}{6} \big/ \frac{7}{8}},\qquad
\frac{\frac{1}{2} \big/ \frac{3}{4}}{\frac{5}{6} \big/ \frac{7}{8}} \bigg/ \frac{\frac{9}{10} \big/ \frac{11}{12}}{\frac{13}{14} \big/ \frac{15}{16}},\qquad\ldots$$ (which relates to the Thue-Morse sequence) can be shown to converge to \(\sqrt{2}/2\).

There are many pretty/unexpected/crazy formulas obtainable from the natural numbers \(1,2,3,4,\ldots\) that could not fit in this post. What are some of your favorites?