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?

2 thoughts on “Astounding “Natural” Identities

  1. my favorite is series on basel problem… 😀
    your blog very interesting, sir..
    do yo have some references about chebyshev’s inequality(integral form)?
    that is my final project topic, and i’m stuck with the material( googling give me too high material that requires functional analysis whereas i want to study “interesting” stuff about inequality like competition problem :D)

    1. ineq,

      I agree that the Basel problem is quite elegant. Thanks!

      Here are two ideas for (hopefully) interesting places to go with Chebyshev’s integral inequality. I hope this is not too late to be helpful!

      One recommendation is Problem 9 from the 1995 Austrian-Polish Mathematics Competition: Prove that for any positive real numbers x, y and any positive integers m, n, $$\begin{gather*}(n-1)(m-1)(x^{m+n}+y^{m+n}) + (m+n-1)(x^m y^n + x^n y^m) \\ \ge mn(x^{m+n-1}y+y^{m+n-1}x).\end{gather*}$$ (The solution in Note of Titu and Harazi on Algebraic Inequalities (page 268) applies the Chebyshev integral inequality to the functions \(t^{n-1}\) and \(t^{m-1}\) on the interval [x,y].)

      Another possibility is to prove (or look up a proof of) a partial strengthening of Chebyshev’s integral inequality by Anderson (later improved by Fink): if f and g are increasing, convex functions on [0,1] with f(0)=g(0)=0, then $$\int_0^1 f(x)\cdot g(x)\,dx \ge \frac{4}{3} \int_0^1 f(x)\,dx \cdot \int_0^1 g(x)\,dx.$$ This can be extended to more than two functions as well; these slides seem to point to useful references.

      Good luck!

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