All Questions
Tagged with pi approximation
131 questions
3
votes
1
answer
121
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+50
Approximation of $\pi$ using an ellipse and a triangle
Here's how my approximation goes:
Take a point on the unit circle (in the first quadrant) and join it to the origin. Now, the area of the sector formed by the line and the x-axis is $\frac{\theta}{2}$....
- 143
6
votes
0
answers
141
views
Approximating $\pi$ as a fraction
Given $n \in \mathbb{N}$, let $p(n) \in \mathbb{N}$ be such that $n/p(n)$ is the best approximation of $\pi$ (denoted as $\tilde{\pi}_n$). I have two main questions:
Is the sequence $\{p(n)\}$ a ...
- 3,585
5
votes
2
answers
179
views
Intuitive explanation for why $\pi$ turns up in $\binom{2m}m \sim \frac{4^m}{\sqrt{m\pi}}$ [closed]
I am aware of the following approximation:
$$\dbinom{2m}m \sim \dfrac{4^m}{\sqrt{m\pi}}$$
Which is equivalent to saying the follwing (as $m \to \infty$):
Given $m$ objects, the number of ways to pick ...
- 881
2
votes
1
answer
88
views
Approximation of $\pi$ by integral and rational number
Via WolframAlpha, I observed that
$$\int_0^1\frac{x^{4n}(1-x^2)}{1+x^2}dx=\frac\pi2-\frac pq \to 0$$
when the integer $n\to\infty$. This gives an approximation of $\pi$ by a rational number. It is not ...
- 14k
0
votes
2
answers
101
views
How to calculate the ratio of convergence for Euler's, Gauss' and Viète's approximation of $\pi$?
Let $\sqrt{6\sum_{k=1}^\infty{\frac{1}{k^2}}}$ be Euler's approximation of $\pi$; $\lim_{n\rightarrow\infty}\frac{2}{g_n}$ Gauss approximation of $\pi$; and $2\cdot\frac{2}{\sqrt{2}}\cdot\frac{2}{\...
- 1
2
votes
1
answer
100
views
Approximating $\pi=4\sum_{n=1}^\infty\frac{(-1)^{n-1}}{2n-1}$ with stable decimal places
Consider the Leibniz formula for $\pi$
$$
\pi=4\sum_{n=1}^\infty\frac{(-1)^{n-1}}{2n-1}.
$$
What is the minimum number of terms needed to calculate $\pi$ accurate to $k$ decimal places, in the sense ...
- 3,585
8
votes
2
answers
284
views
Minimum number of terms to approximate $\pi=4\sum_{n=1}^\infty\frac{(-1)^{n-1}}{2n-1}$
Consider the Leibniz formula for $\pi$
$$
\pi=4\sum_{n=1}^\infty\frac{(-1)^{n-1}}{2n-1}.
$$
What is the minimum number of terms needed to calculate $\pi$ accurate to $k$ decimal places?
My attempt: ...
- 3,585
0
votes
1
answer
186
views
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 ...
- 75
0
votes
0
answers
68
views
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 ...
13
votes
3
answers
597
views
What is the name of this sequence that approximates $\pi$?
I came across this sequence $$\sum_{n=1}^{\infty}\frac{2^n}{2n\choose n}n^m=a_{m}+b_{m}\pi$$ where $a_m$ and $b_m$ are rational with $\lim_{m\to\infty}\frac{a_m}{b_m}=\pi$. I cannot figure out where ...
- 139
22
votes
4
answers
2k
views
Conjecture: The sequence $\frac{2}{n} \sum_{i=1}^n \sqrt{\frac{n}{i-\frac{1}{2}}-1}$ converges to $\pi$
I found that the series
$$s(n) = \frac{2}{n} \cdot \sum_{i=1}^n \sqrt{\frac{n}{i-\frac{1}{2}}-1}$$
converges to $\pi$ as $n \to \infty$.
To verify this I have computed some values:
$n$
$s(n)$
$10^1$
...
- 2,313
-1
votes
4
answers
206
views
Approximations of Euler's constant using $\pi$
Disclaimer: This is for recreational purposes.
Hello MSE! So while my research paper about Euler's constant $\gamma$, I created this amazing approximation for it: $$\frac{\pi^2}{12000}-\ln((10^{-3})!)*...
- 6,707
2
votes
1
answer
146
views
Rational approximation of $\pi$ by recursion
I wonder if the following result is already known and may be considered as interesting : let $\mathcal{C}$ be the real algebra of continuous functions $f:\left[0,1\right]\to\mathbf{R}$, $T:\mathcal{C}\...
- 54
2
votes
5
answers
310
views
Approximations of $\pi$ using radicals
There are many approximations of $\pi$ using trigonometric and rational numbers. But I created this one: $$\pi \approx \sqrt[11]{294204}$$ Which is correct to almost $8$ decimal places. Are there any ...
- 6,707
2
votes
0
answers
222
views
Calculating Pi by subtracting squares from a circumsquare
As you all know, when we inscribe a circle (with radius 1) into a square with side length 2, we cannot approximate the circle's circumference by "folding in" the corners of the square ...
9
votes
3
answers
295
views
What is the strange pattern in the behaviour of this approximation of pi?
I have been playing around with this approximation of pi recently:
$$\lim_{n\to\infty} \sum_{i=0}^{n-1} \frac{n}{n^2+i^2} = \frac{\pi}{4}$$
and although I am perfectly aware that as far as ...
- 91
0
votes
0
answers
75
views
Estimating Pi by Throwing Bread
I remember hearing about a story in which an Italian King (hundreds of years ago) drew a circle and randomly threw bread behind his shoulder, and calculated the percent of bread that landed inside the ...
- 3,437
4
votes
2
answers
156
views
Approximating factorial using identity $\frac{1}{x}!\frac{2}{x}!\cdots\cdot\frac{x}{x}!=\frac{ {x}!\cdot(2\pi)^{\frac{x-1}{2}} }{ x^x\cdot\sqrt{x} }$
I created a function that describes the product of the inverse multiples of a factorial
$$ m(x) = \frac{1}{x}!\cdot\frac{2}{x}!\cdot\frac{3}{x}!\cdots\frac{x-1}{x}!\cdot\frac{x}{x}!$$
for some reasons ...
- 1,038
17
votes
4
answers
1k
views
Why is $\frac{7 \cosh(\sqrt 6)}{13}$ near $\pi$?
$\frac{7 \cosh(\sqrt 6)}{13} = 3.1415926822 ...$
$\pi = 3.1415926535 ... $
Why are these numbers so close to each other? Is this just a coincidence?
P.S.
About ten days ago, I saw this question on ...
- 1,545
1
vote
3
answers
83
views
How many digits of $\pi$ are needed to approximate $\pi^n$ correctly to the nearest whole number?
In his piece on $\pi^{\pi^{\pi^\pi}}$, Matt Parker plots a graph showing that if $\pi$ is approximated to $n$ digits, then whenever $m \leq 2 n$ (roughly), $\pi^m$ is correct to the nearest whole ...
- 169
2
votes
2
answers
153
views
Calculating error for $\pi$ in an expansion
I have the following equation: $\quad\pi = 4\arctan(1/2) + 4\arctan(1/3).$
I want to calculate how many terms of the expansion ($n$) I have to calculate in order for the error to be less than $10^{-...
- 515
0
votes
2
answers
84
views
Identifying this approximation of pi
I've come across this approximation of pi, which I'm struggling to put a name on. I want to do further research on it but I can't find any evidence of it on the internet. It is an infinite sum:
$\pi = ...
- 59
2
votes
1
answer
453
views
Strange similar approximations of pi and e
I just discovered these after approximation by Borwein and Bailey for $\pi$
And I'm assuming the latter just converges to e as the power of x increases although I'm not sure how to go about it (even ...
- 266
16
votes
2
answers
655
views
Charming approximation of $\pi$: $2\left(\frac{1}{2}\right)^{\phi/2}+2< \pi$, where $\phi$ is the golden ratio
Prove that :
$$2\left(\frac{1}{2}\right)^{\phi/2}+2< \pi$$
where $\phi:=\frac12(1+\sqrt{5})=1.618\ldots$ is the golden ratio.
How I came across this approximation?
Well, I was studying the ...
- 3,742
23
votes
2
answers
2k
views
Why this approximation for $\pi$ is so accurate?
Berggren and Borwein brothers in "Pi: A Source Book" showed a mysterious approximation for $\pi$ with astonishingly high accuracy:
$$ \left(\frac{1}{10^5}\sum_{n={-\infty}}^\infty e^{-n^2/10^{10}}\...
- 351
14
votes
0
answers
333
views
Why Is $\ln 23+\cfrac{1}{\color{red}{163}+\cfrac{1}{1+\cfrac{1}{\color{red}{41}}}}\approx\pi$
I know from reading that the Heegner number 163 yields the prime generating or Euler Lucky Number 41. Now apparently $\ln23<\pi$ and this can be shown without calculators. I noticed that
$$
\pi-\...
- 3,938
-2
votes
1
answer
56
views
Is it possible :$\pi \sim{\frac{-\gamma}{(\sqrt{2}e^2-\gamma)^4}}-e^{-13}+2(17^{\frac13}-1)-2\phi^{-16}-\sum_{n=1}^{\infty}(\frac{1}{17})^{2n+3}$?
I have done many attempts to give another approximation for $\pi$ I have got this
$$\pi \sim{\frac{-\gamma}{(\sqrt{2}e^2-\gamma)^4}}-e^{-13}+2(17^{\frac13}-1)-2\phi^{-16}-\sum_{n=1}^{\infty}(\frac{...
- 7,123
6
votes
1
answer
162
views
Succinct proof that $\frac\pi4+\frac\pi6+\log2\gt2$
In answering Average angle between two randomly chosen vectors in a unit square, I noticed that the average angle formed by two vectors uniformly picked in the unit square, $\frac\pi4+\log2-1\approx0....
- 241k
3
votes
0
answers
99
views
Using four integers from $1$ to $20$ to approximate $\pi$
How close can you get to $\pi$ (as close as possible) using these rules?
1) You can only use at most four integers ranging from $1$ to $20$, each only once.
2) You can only use plus, minus, ...
- 143
21
votes
5
answers
823
views
Happy $\pi$-day! Is it true that $\sum_{p \;\text{prime} } \frac{1}{{\pi}^p} < \pi -\lfloor \pi \rfloor$?
Today is a $\pi$-day and I made this exercise for that purpose (and not only for that!):
Let: $$\phi = \sum_{p \;\text{prime} } \frac{1}{{\pi}^p}$$
By applying only knowledge of calculus and, ...
user757601
0
votes
1
answer
95
views
Why does one approximation of pi equation decrease error, while the other increases it with more iterations?
When approximating pi, these two equations give radically diffrent relative errors? is this due to floating point arithmetic or the equations themselves?
They remain relatively equal for the first 14 ...
- 1
3
votes
0
answers
185
views
Approximation of $|\phi^{\pi}-\pi^{\phi}|$
Show that $$|\phi^{\pi}-\pi^{\phi}|\leq \operatorname{T},$$
where $\phi$ is the golden ratio and $\operatorname{T}$ the Tribonacci Constant
Using a calculator, we have $|\phi^{\pi}-\pi^{\phi}|=1....
- 3,742
1
vote
2
answers
259
views
Rational approximations for $\pi$ using Fibonacci numbers?
It is (well?) known that
$$\frac{\pi} 4 = \sum_{k=1}^\infty \arctan \left ( \frac1{F_{2k+1}} \right )$$
Where $F_k$ denotes the $k$-th Fibonacci number. However, any truncation of this sum is ...
4
votes
1
answer
516
views
How does this iterative method of calculating $\pi$ work?
I found some code online that calculates $\pi$ to an arbitrary number of decimal places. I don't understand why the calculations work or how to find more information about the method used to derive ...
- 143
2
votes
0
answers
134
views
Approximation of the basel problem
Why is it, that $$\sum_{n=1}^{x} \frac1{n^2} $$ is about equal to $(1+\frac1{2x})^{x-1}$? Is it possible to give an error term? My observation is, that for $x=1,2,3$ the two terms are equal but with $...
- 40
0
votes
0
answers
146
views
recursive formula for pi stumbled upon
Looking for a function that would take a long time to test some multiprocess stuff, I calculated x'=sin(x)+x . To my surprise the values tend towards pi after only a few iterations, for all starting ...
- 111
11
votes
1
answer
263
views
Strange approximation to $\sqrt{\pi}$
Let
$$\alpha = \sqrt{2\sin^2 1+\sqrt{2\sin^2 2 + \sqrt{2\sin^2 3 + \cdots}}} =\sqrt{3.1415...}$$
Prove that $\alpha^2 \neq \pi$. It is a remarkable approximation though.
- 3,693
2
votes
0
answers
75
views
A crude approximation to $\pi$ using Gamma function
Recently, I found using my calculator a weird result:
$$
\label{eq:1}\tag{1}
G_{\text{II}}=\Gamma(\Gamma(\pi +1)+1) \approx 7380\frac {5}{9}\text { or }\frac {66425}{9}
$$
Here $\Gamma(z)$ is the ...
- 1,240
1
vote
0
answers
118
views
Are these $\pi$ approximating algorithms known?
I was fiddling around with inscribed and circumscribed n-gons of a circle, and using a couple $\sin$ and $\tan$ formulas I came up with two algorithms that approximate $\pi$ pretty fast.
The first ...
- 2,807
0
votes
1
answer
64
views
How do $i$ and $\pi$ show up in this result from a Google search?
I am just curious and play a bit here. I googled (-1e-10)^(-1e-10) and got
(-1e-10)^(-1e-10) =
$1 - 3.14159266 × 10^{-10} i$
I guess this is some form of ...
- 157
4
votes
4
answers
327
views
Show that $\left\vert\frac{\pi}{4} - \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9}\right)\right\vert < 0.1$
Show that $$\left\vert\frac{\pi}{4} - \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9}\right)\right\vert < 0.1 .$$
I know that $\arctan 1 = \frac{\pi}{4}$ and that the sequence ...
3
votes
2
answers
314
views
Limit of sum of areas of infinite amount of triangles
I apologize for the possible incorrect use of math terms since English is not my native language and I'm not a mathematician, but this issue came to my mind about a month ago and I was unable to solve ...
8
votes
0
answers
422
views
Is there a integer that makes $e^{\pi\sqrt{n}}$ closer to an integer than 163?
$e^{π\sqrt{163}}$ is almost an integer about $262537412640768744$.
Let $\delta = -\log_{10}{\left|[x] -x\right|}$, where $[x]$ means round $x$.
$\delta(e^{π\sqrt{163}}) \approx 12.125$, I searched ...
- 1,270
1
vote
2
answers
277
views
Optimal Fixed-Digit Rational Approximations of $\pi$
Is there a systematic way of finding optimal rational approximations to $\pi$ whose numerator and denominator have at most $n$ digits?
More precisely:
Let $D_n$ be the set of all positive integers ...
- 4,815
0
votes
1
answer
87
views
How to use infinite series to bound $\pi$.
Given that:
$\pi = \sum_{k=0}^{\infty} \frac{1}{16^k}\left(\frac{4}{8k+1} - \frac{2}{8k+4}-\frac{1}{8k+5}-\frac{1}{8k+6}\right)$ and $0 \le\left(\frac{4}{8k+1} - \frac{2}{8k+4}-\frac{1}{8k+5}-\frac{1}...
- 75
0
votes
2
answers
1k
views
Definition of the approx. symbol
Take an unending number, say e.g $π$. If we want to show $π$'s value, should we use the approximately notation or equal sign when writing:
$π = 3.14...$ or $π ≈ 3.14...$
This might be a really ...
- 1,064
3
votes
1
answer
235
views
Nice result that I can't prove: $\int_{-2}^{2} \tan^{-1} \bigg( \exp(-x²\text{erf}(x)) \bigg) \;dx=\pi$
I'm always trying to find the integral representation of $\pi$ using some interesting special function, at this time I have got the below representation
$$I=\int_{-2}^{2} \tan^{-1} \bigg( \exp(-x^2\...
- 7,123
0
votes
0
answers
279
views
How many terms it takes for the Leibniz series to converge to three decimal places of accuracy?
I need to find out how many terms it takes for the this series to converge to three decimal places of accuracy of Pi. e.i how many it terms it takes to obtain the value 3.141 from, the series: Leibniz ...
- 3
3
votes
1
answer
1k
views
Approximating pi rate of convergence
I have been reading about a method for approximating $\pi$ using two uniform distributions and the ratio of points that lie within the circle compared to the square formed by the two uniform ...
- 35
0
votes
2
answers
138
views
Is the solution to $\frac{e^n}{n} = e^2$ related to $\pi$?
I learned recently that the solution to $\frac{e^n}{n}=e^2$ is 3.146.., which is very near $\pi$. Is this just a coincidence, or is there something to the expression that leads to it being so close to ...
