# 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. ineq says:

my favorite is series on basel problem… 😀
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)
thanks

1. Zachary Abel says:

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!