What is the largest $n$ such that each $n$-digit number appears in the...
As the title of the question says. Of course everybody believes that $\pi$ is normal, so that the question 'what is the largest $n$ such that every $n$-digit number appears in the digits of $\pi$' is...
View ArticleDoes the sum $\sum_{n=1}^{\infty}\frac{\tan n}{n^2}$ converge?
Does the sum $$\sum_{n=1}^{\infty}\frac{\tan n}{n^2}$$converge?
View ArticleDoes convergence of the Flint-Hills series imply convergence of the...
The Wolfram MathWorld page on the Cookson Hills series states:It is not known if this series converges since $\sec^2 n$ can have sporadic large values.The series is$$\sum_{n=1}^{\infty}\frac{\sec^2...
View ArticleWhy isn't the lateral surface area of a cone equal to $\pi RH$?
Proof of the surface area of a cone doesn't make sense (to me at least).The surface area of a cone is$$\pi R G + \pi R^2 \tag1$$where $R$ is the radius of the base, and $G$ is the cone's generatrix....
View ArticleIntegral involving elliptic integral functions
Recently I came across this identity:$$\int _0^1\:\frac{K\left(x\right)}{1+x}dx=\frac{\pi ^2}{8}$$Where:$$K\left(x\right)=\int _0^{\frac{\pi }{2}}\:\frac{1}{\sqrt{1-\left(x\sin \left(\theta...
View ArticleA continued fraction for $\frac{3\sqrt{3}}{\pi}$
I found the following continued fraction identity:$$\frac{3\sqrt3}{\pi}=1+\cfrac{1}{1+\cfrac{2}{2+\cfrac{3}{1+\cfrac{4}{3+\cfrac{5}{1+\cfrac{6}{\ddots}}}}}}.$$The partial numerators...
View ArticleIs there an integral that proves $\pi > 333/106$?
The following integral,$$ \int_0^1 \frac{x^4(1-x)^4}{x^2 + 1} \mathrm{d}x = \frac{22}{7} - \pi $$is clearly positive, which proves that $\pi < 22/7$.Is there a similar integral which proves $\pi...
View ArticleRepeating digits in pi
Running an experiment to see how many times different integers of different lengths repeat themselves in the first million digits of pi(searched number: how many times it appears)When searching 1...
View ArticleIs there a known geometric framework viewing mixed-radix numeration,...
I'm looking for references (or a name) for a picture that seems to unify several classical objects. I'd like to know whether it already exists in the literature, and if so under what name.The picture....
View ArticleWhy is the expected value of this random nested radical equal to $4/\pi$?
To celebrate Pi Day, I took a look at some classic identities for $\pi$ again, particularly the nested roots associated with Viète’s formula. Such as this one:$$\pi=\lim_{k \to\infty} 2^k...
View ArticleA continued fraction for Baxter's four-coloring constant
I found the following infinite continued fraction:$$\operatorname{C_{B4CC}}=\cfrac{2}{1+\cfrac{2}{1+\cfrac{6}{1+\cfrac{3}{1+\cfrac{10}{1+\cfrac{4}{\ddots}}}}}}$$where $\operatorname{C_{B4CC}}$ denotes...
View ArticleDistribution of the digits of $\pi$
Can anything be stated about the distribution of the digits of $\pi$, i.e., if I were to sample $n$ digits of $\pi$, can anything be said about the probability to observe certain digits, or is there...
View ArticleA continued fraction for the reciprocal of Gauss's constant
I found the following infinite continued fraction:$$\frac{1}{G} = \frac1{(2\pi)^{-3/2}\,\Gamma^2(1/4)} = \cfrac{4}{1+\cfrac{5}{1+\cfrac{6}{1+\cfrac{9}{1+\cfrac{8}{1+\cfrac{13}{\ddots}}}}}}$$where $G$...
View ArticleIs it mathematically correct to write $e^{i\pi}=i^2$?
I know that Euler's identity gives $e^{i\pi}=-1$, and since $i^2=-1$, it follows that $e^{i\pi}=i^2$.My questions are:Is it mathematically correct to write $e^{i\pi}=i^2$, or is this considered poor...
View ArticleProving $\pi^3 \gt 31$
$$\large \pi^3 \gt 31$$Using a calculator, $\pi^3/31 \approx 1.0002$, so I thought this may be challenging to do by hand.It is extremely easy with the use of any calculator, so I was wondering now:Can...
View ArticleAbout Legendre's like relations generalizations.
ContextBeing:$$K(k)=\int_{0}^{\pi/2}\frac{dt}{\sqrt{1-k^2\sin^2{t}}}=\frac{\pi}{2}\sum_{n=0}^{\infty}\frac{(2n)!^2k^{2n}}{2^{4n}n!^4},\hspace{.5cm}k...
View ArticleDoes every digit in base $n$ appear in the expansion of $\pi$? [duplicate]
So I was thinking of the conjecture that every finite combination of digits appears somewhere in the expansion of $\pi$. This led me to a wonder:For every integer base $n \ge 2$, does every digit...
View ArticleWhat is the probability of a specific sequence of 11 digits occurring in a...
This isn't homework. I'm actually wondering how likely it is that any particular 11-digit telephone number will occur in the first billion digits of $\pi$. My probability course was way too long ago,...
View ArticleOn Ramanujan's fastest series.
ContextWith some effort we can show that Ramanujan's fastest series implies:\begin{align}\frac{8E(k_{58})K(k_{58})}{\pi^2}-\frac{aK^2(k_{58})}{\pi^2}=\frac{\sqrt{58}}{29\pi},\tag{1}\end{align}with:...
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