Questions tagged [pi]
The number $\pi$ is the ratio of a circle's circumference to its diameter. Understanding its various properties and computing its numerical value drove the study of much mathematics throughout history. Questions regarding this special number and its properties fit in here.
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Does a closed-form expression exist for $ \int_0^\infty \ln(x) \operatorname{sech}(x)^n dx $?
I am trying to find a closed-form expression for the following integral
$$
\int_0^\infty \ln(x) \operatorname{sech}(x)^n dx
$$
There are specific values that I would like to generate
(Table of ...
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Find a simple proof that π is irrational
I know there are many questions on the site about finding a proof that π is irrational, but I'm posting the question separately to discuss a particular proof further
We know that the Wallis Product is ...
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What is the limit $\lim_{n\to \infty} \cos(n)^{n^2} $
Two years ago I found this question and wasn't able to make the slightest advancement.
$\lim_{n\to \infty} \cos(n)^{(n²)} $
The question is does the limit exist, where n is natural only (of course it ...
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Show that $\pi$ is not algebraic over the field $\mathbb A$ of algebraic numbers.
This is one of the exercises in my abstract algebra book (Nicholson) and it's just the title:
Show that $\pi$ is not algebraic over the field $\mathbb A$ of algebraic numbers.
All I know what to do ...
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How do I calculate the digit number n of pi?
I want to create a pi calculator with no limit to precision, but I cannot find a formula for finding only the digit number n of pi.
Here is what I mean:
Let's say I want to find the 4th digit of pi. ...
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Euler's proof of $\frac{\pi}{6}=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\cdots$
Euler proved
$$\frac{\pi}{6}=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\cdots$$
where the reasoning of the signs thus is prepared, so that of the second may be had as $-$, prime ...
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Approximating $\pi=4\sum_{k=1}^\infty \frac{(-1)^{k+1}}{2k-1}$ [duplicate]
Consider the series
$$
\pi=4\sum_{k=1}^\infty \frac{(-1)^{k+1}}{2k-1}
$$
How many terms of this series do I need to consider to have an approximation of $\pi$ accurate up to $10$ decimal places (for ...
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Identity involving the number pi [duplicate]
Why is this identity true?
$\pi$ = $\lim_{n\to\infty} 2^n\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+...}}}} $
Where the number of two's inside the big square root is equal to n.
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Conjecture: In Pascal's triangle with $n$ rows, the proportion of numbers less than the centre number approaches $e^{-1/\pi}$ as $n\to\infty$.
Consider Pascal's triangle with $30$ rows (the top $1$ is the $0$th row). The centre number is the number in the middle of row $30\times \frac23=20$, which is $\binom{20}{10}=184756$. The proportion ...
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Deriving the continued fraction for Pi [closed]
So I was searching online for methods to approximate Pi and found this
continued fraction that supposedly approximates to Pi when continued infinitely. I've tried searching all over the internet for ...
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Found a relation regarding the primes, is this interesting?
Define $S_{odd}$ as all $n\in N $ where $n$ is the product of an odd number of distinct primes. Define $S_{even}$ similarly. Thus:
$$S_{odd} = \{2,3,5,...,30,42,....\}$$
$$S_{even} = \{6,10,14, ....,...
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Is there a self-correcting iterative method for approximating pi without using transcendental functions?
The Newton-Raphson method is an iterative method for finding a root of a function, and it is self-correcting in the sense that any error in the initial input is reduced with each iteration so that it ...
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For what $n$ is the sequence consisting of the first $n$ digits of $\pi$ a palindrome
Can we find sequences of the first $n$ digits of $\pi$ so that this sequence is a palindrome, i.e $ 3,1,4, ...., 4,1,3$. Trivially, this is the case for $n=1$.
I say the following statement:
There are ...
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Using Ramanujan-type series for $1/\pi^m$ to find formulas for $\zeta(2),\, \zeta(3),\, \zeta(4)$?
As described by Guillera in "Ramanujan Series with a Shift", one nice thing about Ramanujan-type $1/\pi^m$ formulas is by "shifting" them, they can yield a second value which may ...
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Why $1+\frac{1}{3}-\frac{1}{5}-\frac{1}{7}+\cdots=\frac{\pi}{2\sqrt{2}}$? [duplicate]
Wikipedia's List of formulae involving $\pi$ claims
$$\sum_{n=0}^\infty \frac{(-1)^{(n^2-n)/2}}{2n+1}=1+\frac{1}{3}-\frac{1}{5}-\frac{1}{7}+\cdots=\frac{\pi}{2\sqrt{2}}$$
and cites Chrystal's Algebra, ...
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Series $\sum_{n=1}^{\infty}\left(\frac{60n^{2}-43n+8}{n^{4}}\right)\binom{4n-2}{2n-1}\binom{2n}{n}^{-4}=2\zeta(2)$
Recently through Integer Relation Algorithms I was able to find the following Series For $\pi$ :
$$\sum_{n=1}^{\infty}\left(\frac{60n^{2}-43n+8}{n^{4}}\right)\binom{4n-2}{2n-1}\binom{2n}{n}^{-4}=2\...
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Is $\sqrt{\pi}=2\sum_{k=0}^{\infty} \frac{e^{-k^2}-e^{-(k+1)^2}}{2k+1}$?
Is it true that
$$
\sqrt{\frac{\pi}{4}}=\sum_{k=0}^{\infty} \frac{e^{-k^2}-e^{-(k+1)^2}}{2k+1}
$$
Context: Attempting to find an easier proof for this estimate, when $x=1$.
My attempt: Leibniz's ...
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The closed-form of $1-5\left(\frac{1}{2}\right)^k+9\left(\frac{1\cdot3}{2\cdot4}\right)^k-13\left(\frac{1\cdot3\cdot5}{2\cdot4\cdot6}\right)^k+\dots$?
(A related MSE question by P. Singh.) First define,
$$F_k = 1-5\left(\frac{1}{2}\right)^k+9\left(\frac{1\cdot 3}{2\cdot 4}\right)^k-13\left(\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\right)^k+17\left(\...
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How is $f_1(n)$ not computable but $f_2(n)$ is?
I came across these two introductory examples on the topic of computable.
$$f_1(n) = \begin{array}{cc}
\Bigg \{ &
\begin{array}{cc}
1 & ,\text{if n appears in the decimal ...
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$\pi$ approximation method confusion
I am reading a book (A History of Pi) in it there is a story about how Indian mathematicians found the value of $\pi$ by inscribing the polygons in a circle with diameter of 100 and doubling the sides ...
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A cool integral: $\int^{\ln{\phi}}_{0}\ln\left(e^{x}-e^{-x}\right)dx=-\frac{\pi^2}{20}$
I was looking at the equation $\ln{e^{x}-e^{-x}}$ and found that the zero was at $x=\ln{\phi}$ where $\phi$ is the golden ratio. I thought that was pretty cool so I attempted to find the integral. I ...
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Can we estimate the difference between $S_n-\pi$ where $S_n$ is the truncated BBP formula for $\pi$?
The truncated BBP formula for $\pi$ is
\begin{align*}
S_{n}=\sum_{k=0}^{n}\left( \frac{4}{16^k(8k+1)}
-\frac{2}{16^k(8k+4)}
-\frac{1}{16^k(8k+5)}
-\frac{1}{16^k(8k+6)}\right)
\end{align*}
where $\lim_{...
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Probability that at least 5 adjacent length-4 blocks of a certain kind occur in a string of 26 random digits
Let a "$(4,2)$-block" be any string of four digits exactly two of which are equal.
Question: What is the probability that in a string of $26$ decimal digits, chosen independently and ...
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Euler's totient function to estimate $\pi$
Euler's totient function $\varphi(n)$ to estimate $\pi$
$$\pi = \sqrt{6 \times \left( \lim_{n \to \infty} \left( \frac{1}{n} \sum_{i=1}^{n} \frac{\varphi(i)}{i} \right)^{-1} \right)}$$
The Idea
...
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Approximate $\pi$ using Gauss Circle Problem and Pick's Theorem.
Introduction
For any non-negative integer $k$, $x^2+y^2=5^k$ has $4(k+1)$ integer solutions (vertices).
For example, here are 3 circles for $k=1,2,3$. The blue dots represent the vertices (integer ...
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Verify this proof of the Leibniz $\pi$ formula
Consider $F(x)=\tan(x)-1$
The roots of the equation are $\dfrac{\pi}{4}+n\pi$ i.e
$(\dfrac{\pi}{4},\dfrac{5\pi}{4},\dfrac{9\pi}{4},\cdots)$
and
$(\dfrac{-3\pi}{4},\dfrac{-7\pi}{4}, \dfrac{-11\pi}{4},\...
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Is there a known explanation for the Feynman point?
The Feynman point is a mathematical coincidence. It states that from position 762, there are six consecutive nines in the decimal expansion of pi. Some mathematical coincidences have an explanation, ...
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Derivation of the Wallis' product using trigonometry
I wanted to know if the method I used is valid and not just pure luck:
Consider the function:
$$\dfrac{\cos(\frac{\pi x}{2})}{(x^2-1)}$$
The roots of the equation are$\quad\pm3,5,7,9,\cdots$
So the ...
user1211726
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Possible connection between prime numbers in binary and $\pi$
I just realized I asked a question with a very similar title a while back. This is not a duplicate. It is another conjecture though.
Conjecture: $\lim\limits_{x\to\infty}\mu \{ f(2), f(3), f(5), ... ,...
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Need to simplify formula $ \pi \sim \frac{8}{n^2} \cdot \sum\limits_{i=0}^{n} \sqrt{i \cdot (n - i)} $
This formula can be simplified?
$$ \pi \sim \frac{8}{n^2} \cdot \sum\limits_{i=0}^{n} \sqrt{i \cdot (n - i)} $$
I am trying to find an alternative formula for circle area.
I stacked to analyze this ...
user1206215
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How to define $\pi$ from geometry to the limit of a sequence?
There are several definitions of $\pi$ based on the limit of some sequences or series. (Maybe) The most famous example is the solution of the Basel problem:
$$\pi = \sqrt{6 \sum_{n=1}^{+\infty}\frac{1}...
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Non trivial $f$ such that $\int_0^1 x^n f(x) dx=a_n\pi+b_n$
I need a non trivial function $f(x)$ such that $$\int_0^1 x^n f(x) dx=a_n\pi+b_n$$ where $a_n,b_n\in\mathbb{Z}$ and $n\in\mathbb{N}$
We know that $$\pi=4\int_0^1 \sqrt{1-x^2}\ dx$$
By Binomial ...
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Archimedes: Inscribed regular polygon smaller perimeter than a circle?
When Archimedes found the upper and lower boundary for the value of pi, he used an $\color{red}{\textrm{inscribed regular polygon}}$ and a $\color{blue}{\textrm{encapsulating outer regular polygon}}$. ...
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Does pi's decimal expansion certainly repeat once (of course not infinitely many times)??
My 8-year-old asked me this and I could see arguments for either answer.
Is pi guaranteed to have an expansion of the form
3.NNM
where N is a finite sequence of ...
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A sequence derived from $\lceil \sin (2n) \rceil$ with some interesting features
I was playing with Excel and created a sequence with some interesting features.
In column A, list the sequence $\color{red}{a_n=\lceil \sin (2n) \rceil}$ (using the ceiling function).
In column B, ...
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Approximating Pi with fractional log base. Coincidence?
In an attempt to understand logarithms better for an unrelated problem, I stumbled across the value of $\log_{9}(1000)$ is equal to approximately 3.143. I thought this was interesting, so tried to get ...
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Why does the exponent rule [If $a^b = a^c,$ then $b=c$] not apply to imaginary numbers [duplicate]
So I came across this video: https://www.youtube.com/watch?v=R476CTKUIr4
in which the creator shows an incorrect proof of π = 0 and the mistake made. The video proves this using the exponent rule $(a^...
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Using Taylor Series to approximate pi without using arctan?
Whenever I search for ways to approximate pi using Taylor/MacLaurin Series, the example that I always see utilizes the fact that $\tan{\frac{\pi}{4}=1}$. However, I vaguely remember coming across a ...
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Rationality of artan(2/pi)
I was messing around on Wolfram Alpha and I saw that the conversion of arctan(2/pi) from radians to degrees was 32.48. Wolfram Alpha usually shows more decimal spaces, so I was wondering where this is ...
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Does $\sin^{-1}𝑥$ abide with the trig exponent notation?
So I know that the difference between "$\sin^a𝑥$" and "$\sin𝑥^a$" is that "$\sin^a𝑥$" means "$(\sin𝑥)^a$" and "$\sin𝑥^a$" means "$\sin(𝑥^a)$...
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Exponential Function with big exponents
Does somebody know how I can simplify something like $e^{2022*i*\pi/4}$ or $e^{2022*i*\pi/8}$ by hand? I tried to image it on the unit circle. Without the 4 or 8 it would be easy since 2022 is a ...
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Representing logical operations in math/logical notation.
I want to use notation like this for loop (which was used in an answer to this question) for some code I have written to calculate the "solar time" of a timestamp given it's location and ...
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Limit involving $\pi$ and the Fibonacci sequence. [duplicate]
In this blog I found a curious limit that involves the Fibonnaci sequence and the number
$ \pi$.
$$\lim\limits_{x\to \infty} \frac{\sum\limits_{i=0}^{x}log F_i}{log (lcm \left \{ F_i \right \}_{i=0}^{...
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Simplifying a double summation related to a Ramanujan-type pi formula $1/\pi$?
I came across an integer sequence for a Ramanujan-type formula for $1/\pi$, namely,
$$a(n)=\color{brown}{\sum_{k=0}^n \binom{n}{k}\binom{2n}{n}\binom{2k}{k}^{-1}}\sum_{j=0}^k\binom{k}{j}^4 = 1, 4, 36, ...
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Estimate $\pi$ using a 3D Random Walk.
A 3D Random Walk according to WolframAlpha says:
$P(\text{{probability of eventual return to origin}}) = 1 - \frac{{16 \sqrt{{\frac{2}{3}}} \pi^3}}{{\Gamma(\frac{1}{24}) \Gamma(\frac{5}{24}) \Gamma(\...
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Proving $\sum_{k=1}^\infty\left(\frac{(n^k+1)}{(n+1)^k}\zeta(k+1)\right)=\psi(\frac{n}{n+1})-\psi(\frac{1}{n+1})=\pi\cot(\frac{\pi}{n+1})$
I'm an amateur/hobbyist mathematician, and I found this interesting relationship about 6 years ago, but haven't ever quite understood it! I feel like this is related to how Digamma and Zeta are ...
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Is the limit of this sequence indeed 2 over Pi? [duplicate]
Sequences $(a_n), (b_n)$ are defined as follows:
\begin{align*}
a_0 = 0;\quad a_{n+1} = a_n + \sqrt{a_n^2+1};
\quad b_n = \frac{a_n}{2^n}; \quad n \geq 0.
\end{align*}
It is not difficult to prove ...
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Show $\pi ≈ \frac{4\sum_{i=0}^\infty\left(\left\lfloor \frac{5^k}{4i+1}\right\rfloor-\left\lfloor\frac{5^k}{4i+3}\right\rfloor\right)-2(k+1)}{5^k}$
Lemma
For any non-negative integer $k$, $x^2+y^2=5^k$ has $4(k+1)$ integer solutions.
Pick's Theorem
Suppose that a polygon has integer coordinates for all of its vertices. Let $i$ be the number of ...
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Prove $\lim_{n\to\infty}\int_0^{a} \left(\sqrt{2n/\pi-x^2}-\sqrt{2n/\pi-a^2}\right)dx=1/6$ where $a$ is the largest real root of $4x^6+x^2=2n/\pi$.
I've never seen anything like this before: an unsolvable cubic, within a definite integral, within a limit (which applies to the cubic and the integral), resulting in a simple closed form.
Prove $\...
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Show by hand without any computer that $\frac{\pi510}{\ln\left(510\right)}<257$
My last question of the day (and for a moment) :
There is no real motivation just maths for fun and curiosities. I come across this calculating some number related to the PNT.
Show that :
$$\frac{\...
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