All Questions
Tagged with pi convergence-divergence
31 questions
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Prove that the limits of these two recurring sequences is pi.
I'm trying to prove that these two sequences $a_n$ and $b_n$ converge to $\pi$, but cannot find a method of doing so. The two sequences are defined as such:
$a_0=2\sqrt3, b_0=3$
$a_n= \frac{2a_{n-1}b_{...
- 147
1
vote
0
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74
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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 ...
- 3,585
3
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1
answer
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How to prove absolute and total convergence of $\sum_{n=1}^{+\infty}(-1)^n\frac{\pi^{nx}}{\pi^{nx^2}+\sqrt{n}}$?
Consider the series of functions
$$\sum_{n=1}^{+\infty}(-1)^n\frac{\pi^{nx}}{\pi^{nx^2}+\sqrt{n}}, \qquad x\in\mathbb{R}.$$
I want to study for which $x\in\mathbb{R}$ the series converges.
I start ...
user603537
2
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1
answer
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How much we know about decimals of irrational numbers? Test of convergence for an alternating series of digits
This is an imaginary exercise that my math teacher gave me to meditate:
For any real number $(*)$ $r=\overline{r_0.r_1r_2...}$, expressed using $10$-base decimals, define the following sum:
$$S(r):=\...
- 921
1
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0
answers
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convergence or divergence of the series $\sum_{n=1}^\infty \frac{ 1}{n^3 \sin^2(n)}$ and similar series
In one exercise of the Calculus textbook, Thomas' Calculus, 13th edition, there is an exercise (Section 10.4, 71) says it is not known yet if the series
$$\sum_{n=1}^\infty \frac{1}{n^3 \sin^2(n)} $$
...
- 1,673
43
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1
answer
873
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A strange occurrence in the decimal digits of $\pi$?
I was messing around with various ways to calculate $\pi$ with my computer, and I noticed something a bit strange. I was using $\frac{1}{1}-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+... = \frac{\pi}{4}$, ...
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Proving $\frac 12+\sum\limits_{n=0}^\infty\frac{(-1)^n}{(5+6n)(7+6n)}=\frac{\pi}6$ without Leibniz
Show that $$\frac 12+\sum_{n=0}^\infty\frac{(-1)^n}{(5+6n)(7+6n)}=\frac{\pi}6$$
My proof uses the Leibniz series, $$1-\frac 13+\frac 15-\frac 17+\frac 19-\frac 1{11}+\cdots = \frac{\pi}4$$
\begin{...
- 9,595
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1
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Can the radius of convergence be equal to 1 (arctangent function)?
I wanted to find a proof of the Leibniz $\pi$ formula, knowing that $\arctan(1) = \pi / 4.$ As such, I simply needed to find the Maclaurin Series of $\arctan$ and its rate of convergence. I want to ...
- 447
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1
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38
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Proving limit of a sequence without Taylor series
I'm trying to prove that pi is irrational with Niven's proof. In the proof I need to show the following: $lim_{n\to\infty}\frac{\pi^{n+1}a^n}{n!}=0$ where $\pi=\frac{a}{b}$ to obtain the contradiction....
- 105
1
vote
1
answer
179
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I seem to have discovered a new converging series for $\pi$
Recently, I was experimenting with formulas involving circles and convergent series and came across a new type of series which seems to converge to $\pi$ extremely fast. I have yet to see this ...
- 21
6
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1
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271
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Why does Brent/Salamin algorithm double the digits of $\pi$ with each iteration?
The Brent-Salamin-Formula uses the arithmetic-geometric mean to calculate $\pi$.
There are many sophisticated proofs proving very sharp error bounds, for example in Salamin's 1976 paper or in this ...
- 765
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1
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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
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3
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What is the formula for pi used in the Python decimal library?
(Don't be alarmed by the title; this is a question about mathematics, not programming.)
In the documentation for the decimal module in the Python Standard Library, ...
- 41.9k
5
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1
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Help understanding the cause of this pattern when writing π as an infinite series with double factorials
I made a post about a year and a half ago: $\pi$ as an Infinite Series using Taylor Expansion on Equation of a Circle
where essentially I used the Taylor series expansion on $\ y = \sqrt{r^2-x^2}$ (...
- 321
0
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0
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279
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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
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1
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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
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4
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On a conjecture that $\sum\limits_{n=1}^k\frac{1}{\pi^{1/n}p_n}\stackrel{k\to\infty}{\longrightarrow} 2$.
I have made the following conjecture, and I do not know if this is true.
Conjecture:
\begin{equation*}\sum_{n=1}^k\frac{1}{\pi^{1/n}p_n}\stackrel{k\to\infty}{\longrightarrow}2\verb| such that we ...
- 9,595
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Question about $\pi$ [closed]
I'm interested in the limit as you take the square of the number given by the first $N$ digits of $\pi$ (ignoring the decimal point) & then add to it the first $M$ digits of $\pi$ (ignoring the ...
- 150
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1
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Pi series that converges arbitrarily fast.
The old series for $\pi$ is this alternating series:
$$\pi = 4 \sum_{i=0}^{\infty}\frac{(-1)^i}{2i+1}$$
Now, as already noticed, the series is alternating: adding one term overshoots $\pi$ every time. ...
- 1,141
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2
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797
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Very slowly converging formulas for $\pi$
I'm aware that both Leibniz's formula and Willis' product formula for $\pi$ converges at a logarithmic rate.
In a French paper, this formula was given as: "The slowest and heaviest formula imaginable ...
- 1,292
6
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1
answer
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$\pi$ as an Infinite Series using Taylor Expansion on Equation of a Circle
If we write $\ y = \sqrt{r^2-x^2}$ for the equation of a circle in Cartesian coordinates with radius $r$ and perform a Taylor expansion on this equation, then integrate term by term from $0$ to $r$, ...
- 321
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1
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$ \lim_{n\to\infty}\prod_{i=1}^k\sin(nx_i)\ =\ 0 $
Assume that real numbers $x_1,\dots,x_k$ satisfy
$$
\lim_{n\to\infty}\prod_{i=1}^k\sin(nx_i)\ =\ 0
$$
It implies that one of them is a multiplicity of $\pi$, or not necessarily?
I could not find the ...
- 1,822
2
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1
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258
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Error computing $\pi$ approximation
My book suggests the following exercise.
Which one from the following approximation of $\pi$ minimises the error propagation due to rounding errors?
$$\pi = 4(1 - 1/3 + 1/5 - 1/7 + 1/9 - \ldots)$$...
- 787
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Proof of Leibniz $\pi$ formula
I found the following proof online for Leibniz's formula for $\pi$:
$$\frac{1}{1-y}=1+y+y^2+y^3+\ldots$$
Substitute $y=-x^2$:
$$\frac{1}{1+x^2}=1-x^2+x^4-x^6+\ldots$$
Integrate both sides:
$$\...
- 3,455
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1
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Proving Archimedes Sequences Equal $\pi$.
I encountered the following problem in my text An Introduction to Analysis by William Wade. It was the last problem in the section and has an * next to it. I'm not sure if this indicates a challenge ...
- 2,339
7
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3
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753
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What is the fastest way to $\pi$?
There are many known sequences convergent to $\pi$ with different convergence accelerations. For example both of the following sequences are convergent to $\pi$ when $n$ goes to $\infty$:
(a) $a_n=2^{...
user108850
3
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1
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What would the Chudnovsky Algorithm look like as an inifinite series
I'm interested in generating digits of $\pi$ (I'm programming it in Python) and from my research it seems Chudnovsky algorithm is the fastest. Unfortunately for me, the Wikipedia page only really ...
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1
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How to prove that a series expansion of pi has converged to a certain accuracy?
Wikipedia has a great article on methods for calculating pi with arbitrary precision, using for example Machin's infinite series expansion:
$\frac{\pi}{ 4} = 4$ arccot $5 - $arccot $ 239 $
where
...
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1
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Pi approximation
If $d(a,b)=$ largest $n$ such that $a$ and $b$ agree on all digits upto $n$. Eg. $d(\pi,3.14)=3$, $d(0.1234667,0.1234669)=7$. What is the asymptotics of $d(\pi/4,1-1/3+1/5-1/7+\cdots(\pm)1/m)$ as $...
- 13
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11
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Series that converge to $\pi$ quickly
I know the series, $4-{4\over3}+{4\over5}-{4\over7}...$ converges to $\pi$ but I have heard many people say that while this is a classic example, there are series that converge much faster. Does ...
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