Bizarre Integral $\int_0^1...
Does the following equality hold?$$\int_0^1 \frac{\tan^{-1}{\frac{88\sqrt{21}}{215+36x^2}}}{\sqrt{1-x^2}} \, \text{d}x = \frac{\pi^2}{6}$$The supposed equality holds to 61 decimal places in...
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$\pi$ in arbitrary metric spaces
Whoever finds a norm for which $\pi=42$ is crowned nerd of the day!Can the principle of $\pi$ in euclidean space be generalized to 2-dimensional metric/normed spaces in a reasonable way?For Example,...
View ArticleHow to show $\sum_{n=0}^{\infty}\frac{(6n+1)(2n)!^3}{2^{8n}n!^6}=\frac{4}{\pi}$?
ContextWe continue trying to understand Ramanujan's series for $1/\pi$ so this question has this two important references:"Modular equations and approximations to $π$". By S. Ramanujan....
View ArticleExpression for the Gaussian integral $\int_0^\infty e^{-x^6}\,\mathrm dx$
Let $$I_n=\int_0^\infty e^{-x^n}\,\mathrm dx=\frac1n\Gamma(\tfrac1n).$$ Then, we have$$I_2=\frac1{2\sqrt2}\sqrt{2\pi}$$$$I_4=\frac1{4}\sqrt{\sqrt{2\pi}}\sqrt{2\varpi}$$where $\pi=2\int_0^1\frac{\mathrm...
View ArticleBauer's series for $\frac{1}{\pi}$
Recently, someone asked a question involving the expression$$ \sum_{n=0}^{\infty} (-1)^n (4n+1) \left(\frac{(2n-1)!!}{(2n)!!}\right)^3 $$At first glance, I knew that the expression was the value of a...
View ArticleHow to Evaluate $\sum_{n=0}^{\infty}\frac{(-1)^n(4n+1)(2n)!^3}{2^{6n}n!^6}$
I want to Evaluate$\sum_{n=0}^{\infty}\frac{(-1)^n(4n+1)(2n)!^3}{2^{6n}n!^6}.$I tried from arcsin(x) series and got $\frac{1-z^4}{(1+z^4)^{\frac{2}{3}}}=...
View ArticleIs $\large \frac {\pi}{e}$ rational, irrational, or trandescendal?
Is there an argument for why $\large \frac {\pi}{e}$ is rational, irrational, or trandescendal? Can the quotient of any two transcendental numbers (which are not rational multiples of each other) be...
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Proving that the arithmetic-geometric mean of $1$ and $\sqrt{2}$ is...
I have just read on Wikipedia that in 1799 Gauss proved that $$\color {blue}{AGM(1,\sqrt2)=\frac{\pi}{\varpi},}$$ where $AGM$ is the arithmetic–geometric mean (see the previous link) and $\varpi$ is...
View ArticleGeneralization of Viète's Formula for pi
There is a well-known formula that goes as follows: let $a_1 = \sqrt{2}$, and let $a_n = \sqrt{2+ a_{n-1}}$. Then $$\prod_{i=1}^{\infty} \frac{a_i}{2} = \frac{2}{\pi}$$ My questions are about...
View ArticleApproximation for $\pi$
I just stumbled upon $$ \pi \approx \sqrt{ \frac{9}{5} } + \frac{9}{5} = 3.141640786 $$which is $\delta = 0.0000481330$ different from $\pi$. Although this is a rather crude approximation I wonder if...
View ArticleIs $\pi$ known to be level-2 algebraic?
It's known that $\pi$ is not algebraic i.e. there is no finite "polynomial" $P(x) = \sum_{n=0}^{k} a_n x^{c_n}$ s.t. $P(\pi)=0$ where $a_n, c_n$ are rational.We might instead ask, what about...
View ArticleDo all irrational numbers contain each other? [closed]
From my understanding, both $e$ and $\pi$ are both non-terminating and non-repeating numbers that are infinitely long and they contain every possible sequence of digits. If that is true, does that...
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While working on a Ramanujan's series for $\frac1\pi$, I obtained the double...
View ArticleWhat's the limit of this series? [duplicate]
What's the result of the series?$\displaystyle\sum_{k=1}^{\infty}{\frac{k \cdot 2^k}{\binom{2k}{k}}}$Numerical results suggest the answer is $\pi+3$But how to prove it?LLMs give proofs using...
View ArticleWhy does $\pi$ equal $4 \left(\frac{1}{2}!\right)^2$? [duplicate]
I was messing around with factorials and found out that you can do factorial's of fractions using the gamma function $ \Gamma(z+1) $which I thought was cool, and I was plugging in a couple of numbers,...
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Show that $\zeta(2) = \sum_{n>0} \frac{(-1)^{n+1} (2n+1) \pi^{2n+2}...
$$\zeta(2) = \sum_{n>0} \frac{(-1)^{n+1} (2n+1) \pi^{2n+2}} { (2n+3)!} $$How to show this ?In particular I am intrested in showing it without complex numbers or complex analysis.Do similar...
View ArticleWhy is it so difficult to determine whether or not $\pi^{\pi^{\pi^\pi}}$ is...
I have heard that it is unknown whether or not $\pi^{\pi^{\pi^\pi}}$ is an integer. How can this be? $\pi$ is known to many digits and it seems like only a matter of time on a computer to find the...
View ArticleShow that $\int\limits_0^1...
How can it be shown that$$\lim_{p\to\infty}I(p)= \lim_{p \to \infty}\int^{1}_0 (x^x)^{\scriptscriptstyle {(x^x)^{(x^x)^{(x^x)^{(x^x)^{(x^x)...(p \; times)}}}}}} dx= \frac{\pi^2}{12}$$$I(1)=\int^{1}_0...
View ArticleInfinite series for $\pi$ whose general terms involve $\zeta(n)$
If we let $t=\frac\pi4$ in the Laurent series$$\cot t=\frac1t-\sum_{n=1}^\infty\frac{2\zeta(2n)}{\pi^{2n}}t^{2n-1}$$we get$$\pi=4-\sum_{n=1}^\infty\frac{2\zeta(2n)}{4^{2n-1}}.$$I wonder if there are...
View Article$\pi$-Day Special: Alternative Methods for Proving $\int_{0}^{\infty}...
Happy belated $\pi$-day! :)Prove $$\int_{0}^{\infty} \sqrt{\frac{1}{\sqrt{e^x} } - \frac{1}{e^x}} \,\mathrm dx = \pi.$$Here is the method I used (I am interested to see if there are any other...
View ArticleIs $2^{\sqrt2}\cdot\pi$ known to be transcendental number? [closed]
I know that $2^{\sqrt2}$ and $\pi$ are each transcendental. But is it known that their product or sum are also transcendental?
View ArticleNew Formula for Pi? $\frac{1}{\pi}=\sum_{n=0}^{\infty...
While messing around with the gamma function I have stumbled upon the identity $$\frac{1}{\pi}=\sum_{n=0}^{\infty...
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How to prove this recurrence relation for generalized "rounding up to $\pi$"?
The webpage Rounding Up To $\pi$ defines a certain "rounding up" function by an extremely simple procedure:Beginning with any positive integer $n$, round up to the nearest multiple of $n-1$, then up to...
View ArticleWhy is $\frac{\pi}{\log 13}$ so well approximated by...
I found that $\frac{\pi}{\log(13)}$ is approximated by the fraction $\frac{52378158}{42764081}$ with remarkable precision. Using Wolfram Alpha:$$...
View ArticleShow $\frac{6435\sqrt{2}}{158848} \sum_{n=0}^{\infty}...
A long time ago, while scrolling through a Facebook group called “Advanced Integrals and Series,” I came across the remarkable series below. No solution was ever posted there.$$S =...
View ArticleProve that $\alpha:=\sqrt{\pi}+\sqrt 2 \in \Bbb{C} $ is trancendental over...
We want to prove that $\alpha:=\sqrt{\pi}+\sqrt 2 \in \Bbb{C} $ is trancendental over $\Bbb{Q}$.Attempt. We use proof by contradiction and so assume that $\alpha \in \Bbb{C}$ is algebraic over...
View ArticleCan $\pi$ be defined in a p-adic context?
I am not at all an expert in p-adic analysis, but I was wondering if there is any sensible (or even generally accepted) way to define the number $\pi$ in $\mathbb Q_p$ or $\mathbb C_p$.I think that...
View ArticleHow to show that $\frac{K(k')}{K(k)}=\sqrt{7} \implies...
$$K(k)=\int_{0}^{\pi/2}\frac{dt}{\sqrt{1-k^2\sin^2t}}\tag{1}$$ is thecomplete elliptic of the first kind and $k'=\sqrt{1-k^2}$ thecomplementary modulus. We are interested in...
View ArticleAre there more? $~~ e^{\sqrt{\ln(100\pi)}} =10.999996...\approx11 $
I've discovered a beautiful approximation for $\pi$ that uses only the number $11$ and basic arithmetic and logarithmic operations:$$\frac{11^{\ln 11}}{11 \cdot 11 - 11 - 11 + 11 / 11} \approx...
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