Linked Questions

39
votes
26answers
2k views

What are some surprising appearances of $e$?

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 $...
95
votes
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 ...
34
votes
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 ...
15
votes
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 ...
28
votes
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 ...
31
votes
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 ...
22
votes
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 ...
12
votes
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 ...
8
votes
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 ...
13
votes
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 ...
21
votes
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 ...
7
votes
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\% ...
3
votes
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 ...
0
votes
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}{...
2
votes
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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