# Linked Questions

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I recently came across the following beautiful and seemingly out-of-the-blue appearance of $e$: $E[\xi]=e$, where $\xi$ is a random variable that is defined as follows. It's the minimum number of $... 4answers 21k views ### Motivation for Ramanujan's mysterious$\pi$formula The following formula for$\pi$was discovered by Ramanujan: $$\frac1{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^\infty \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}\!$$ Does anyone know how it works, or ... 5answers 8k views ### Does the number pi have any significance besides being the ratio of a circle's diameter to its circumference? Pi appears a LOT in trigonometry, but only because of its 'circle-significance'. Does pi ever matter in things not concerned with circles? Is its only claim to fame the fact that its irrational and an ... 7answers 7k views ### What are your favorite relations between e and pi? [closed] This question is for a very cool friend of mine. See, he really is interested on how seemingly separate concepts can be connected in such nice ways. He told me that he was losing his love for ... 3answers 1k views ### Is there a meaningful example of probability of$\frac1\pi$? A large portion of combinatorics cases have probabilities of$\frac1e$. Secretary problem is one of such examples. Excluding trivial cases (a variable has a uniform distribution over$(0,\pi)$- what ... 5answers 3k views ### Geometric intuition for$\pi /4 = 1 - 1/3 + 1/5 - \cdots$? Following reading this great post : Interesting and unexpected applications of$\pi$, Vadim's answer reminded me of something an analysis professor had told me when I was an undergrad - that no one ... 4answers 8k views ### Factorial of infinity So, I've read in this article that: $$\zeta'(0) = \log\sqrt\frac 1 {2\pi}$$ And that: $$e^{-\zeta'(0)} = 1\cdot2\cdot3\cdots \infty = \infty! = \sqrt{2\pi}$$ I found this result very strange and ... 3answers 2k views ### On the probability that two positive integers are relatively prime In many of the sources I have consulted about this, the "probability" that two positive integers chosen at random are relatively prime is calculated as the limit as$n \to \infty$of the probability ... 9answers 954 views ### Calculate$\pi$By Hand? All over the internet the only hand equation i found was $$\frac\pi4 = 1 - \frac13 + \frac15 - \frac17+\cdots.$$ But this takes something like a thousand iterations to get to four digits, is there a ... 5answers 2k views ### Why are the trigonometric functions so important? The basic functions of trigonometry,$\sin$and$\cos$, are ubiquitous in mathematics. They were originally conceived from geometry, and so it's not surprising that they consistently show up in ... 1answer 733 views ### Interpretation of$\frac{22}{7}-\pi$Integral and series proofs that$\frac{22}{7}>\pi$We can prove that$\frac{22}{7}$exceeds$\pi$by using Dalzell integral $$\int_0^1 \frac{x^4(1-x)^4}{1+x^2}dx=\frac{22}{7}-\pi$$ or its ... 3answers 423 views ### Visual explanation of$\pi$series definition Can you visually explain why the following is true: $$\frac{\pi}{4} = \sum\limits_{k=0}^\infty \frac{(-1)^k}{2k + 1} = \frac{1}{1}-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}\ldots\approx 78.5\% ... 2answers 474 views ### A theta function around its natural boundary Let q = e^{2\pi i\tau}, if$$\psi(q^2)=\sum_{n=0}^{\infty} q^{n(n+1)}$$is one of ramanujan theta functions,is it possible to evaluate the limit$$\lim_{q\rightarrow 1} (1-q){\psi^2(q^2)}$$In fact ... 2answers 81 views ### Are there any formulas or identities that involves \pi but have no obvious trigonometric interpretation? Every formula that involves \pi has an underlying trigonometric interpretation but it is not usually obvious. I wonder if there are any formula like the Gaussian integral f_{n}(x)=\sqrt{\frac{1}{... 1answer 141 views ### Show that the integral can not exceed \frac{\pi^2}{96} Show that$$ \int_{-\infty}^{\infty}\int_{-x}^{x}\int_{-y}^{y}\int_{-z}^{z}e^{-(x^2+y^2+z^2+w^2)}\dfrac{|zw|}{(1+x^2)(1+y^2)} \,dw \,dz\,dy\,dx\le\frac{\pi^2}{96}$$I am not understanding how$\pi\$ ...

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