For questions related to approximations

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65
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3answers
3k views

Is 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 ...
59
votes
9answers
4k views

Find the average of $\sin^{100} (x)$ in 5 minutes?

I read this quote attributed to VI Arnold. "Who can't calculate the average value of the one hundredth power of the sine function within five minutes, doesn't understand mathematics - even if he ...
55
votes
12answers
7k views

Can the golden ratio accurately be expressed in terms of e and $\pi$

I was playing around with numbers when I noticed that $\sqrt e$ was very somewhat close to $\phi$ And so, I took it upon myself to try to find a way to express the golden ratio in terms of the ...
44
votes
4answers
4k views

Motivation for Ramanujan's mysterious $\pi$ formula

The following formula for $\pi$ was discovered by Ramanujan: $$\frac1{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^\infty \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}\!$$ Does anyone know how it works, or ...
30
votes
7answers
1k views

Pi Estimation using Integers

I ran across this problem in a high school math competition: "You must use the integers $1$ to $9$ and only addition, subtraction, multiplication, division, and exponentiation to approximate the ...
27
votes
4answers
3k views

How much gas does a car use to carry its own gas?

I have always been curious about this one. Since the gas has some weight, the car will have to burn some extra gas to carry it's own fuel around. How can I calculate how much that extra gas is? ...
27
votes
1answer
640 views

A series problem by Knuth

I came across the following problem, known as Knuth's Series which originally was an American Mathematical Monthly problem. Prove that $$\sum_{n=1}^\infty ...
24
votes
3answers
1k views

The right “weigh” to do integrals

Back in the day, before approximation methods like splines became vogue in my line of work, one way of computing the area under an empirically drawn curve was to painstakingly sketch it on a piece of ...
23
votes
5answers
3k views

How do you calculate the decimal expansion of an irrational number?

Just curious, how do you calculate an irrational number? Take $\pi$ for example. Computers have calculated $\pi$ to the millionth digit and beyond. What formula/method do they use to figure this out? ...
23
votes
3answers
500 views

Geometric explanation of $\sqrt 2 + \sqrt 3 \approx \pi$

Just curious, is there a geometry picture explanation to show that $\sqrt 2 + \sqrt 3 $ is close to $ \pi $?
21
votes
9answers
3k views

What is the purpose of Stirling's approximation to a factorial?

Stirling approximation to a factorial is $$ n! \sim \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n. $$ I wonder what benefit can be got from it? From computational perspective (I admit I don't ...
19
votes
2answers
546 views

Maximum of Polynomials in the Unit Circle

Let $z_{1},z_{2},\ldots,z_{n}$ be i.i.d random points in the unit circle ($|z_i|=1$) with uniform distribution of their angles. Consider the random polynomial $P(z)$ given by $$ ...
18
votes
8answers
2k views

Rapid approximation of $\tanh(x)$

This is kind of a signal processing/programming/mathematics crossover question. At the moment it seems more math-related to me, but if the moderators feel it belongs elsewhere please feel free to ...
17
votes
3answers
342 views

Approximating $1/z$ by polynomials

Let $C=\{\mathrm e^{\mathrm it}, 0\le t\le 3\pi/2\}$ and $f(z)=1/z$. By Runge's theorem, there is a sequence of polynomials $p_n(z)$ such that $$\lim_n p_n(z)=f(z)$$ uniformly on $C$. Does anyone ...
16
votes
5answers
1k views

How best to explain the $\sqrt{2\pi n}$ term in Stirling's?

I recently showed my Algorithms class how to bound $\ln n! = \sum \ln n$ by integrals, thereby obtaining the simple factorial approximation $$ e \left(\frac{n}{e}\right)^{n} \leq n! \leq ...
16
votes
5answers
3k views

Approximation of $e^{-x}$

Is there a method to mentally evaluate $e^{-x}$ for $x>0$? Just to have an idea when computing probabilities or anything that is an exponential function of some parameters.
16
votes
3answers
248 views

Approximating $100!$

I participated in an Estimathon (a speed contest of Fermi problems) not long ago. It works as follows. Contestants are given questions and they must give a closed range $[a,b]$ which should contain ...
16
votes
1answer
224 views

Approximate value of a slowly-converging sum of $\sum|\sin n|^n/n$

In this question on Math.SE there appears this sum: $$ S = \sum_{n\geq1}s_n, \qquad s_n = \frac{|\sin n|^n}{n}, $$ which converges very slowly. What methods would you suggest for evaluating it ...
15
votes
6answers
8k views

Simple numerical methods for calculating the digits of $\pi$

Are there any simple methods for calculating the digits of $\pi$? Computers are able to calculate billions of digits, so there must be an algorithm for computing them. Is there a simple algorithm that ...
15
votes
0answers
374 views

On Shanks' quartic approximation $\pi \approx \frac{6}{\sqrt{3502}}\ln(2u)$

In Mathworld's "Pi Approximations", (line 58), Weisstein mentions one by the mathematician Daniel Shanks that differs by a mere $10^{-82}$, $$\pi \approx ...
14
votes
1answer
338 views

A cute approximation for $\cot(2\pi x)$(!?)

Numerical calculations and some theory leads to the suggestion that $$\cot(2\pi x) \rightarrow\frac{1}{2\pi}\sum_r \frac{1}{x-r}$$ where $r$ ranges over all the roots of $B_{2n+1}$ (Bernoulli ...
12
votes
9answers
4k views

How to calculate $e^x$ with a standard calculator

Is there a simple method for calculating the $e^x$ ($x\in\mathbb{R}$) with a basic add/subtract/multiply/divide calculator that converges in reasonable time, preferably without having to memorize ...
12
votes
2answers
2k views

Euler's Approximation of pi.

I recently stumbled across the formula: $$\pi=20\arctan\frac{1}{7}+8\arctan\frac{3}{79}$$ developed by Euler, for approximating pi. I evaluated it to several thousand decimal places and up to that ...
12
votes
1answer
241 views

Showing that $\int_0^\infty x^{-x} \mathrm{d}x \leq 2$.

This integral is very closely related to the sophmores dream that states $$ \int_0^1 x^{-x}\mathrm{d}x = \sum_{n=1}^\infty n^{-n} = 1.27\ldots $$ For example here ...
12
votes
1answer
224 views

Why is integer approximation of a function interesting?

I have recently learnt the following result: Let $g \in \mathbb{R}[x]$ be a polynomial with $g(0) = 0$. Then, for any $\varepsilon > 0$, the set of positive integers $n$, such that $g(n)$ is ...
11
votes
4answers
284 views

Why is this number so close to $1$?

The only positive solution of the equation $\sin (\tan x) = x$ is at a number $a = 0.999906...$. Is it a coincidence that the number $a$ is so close to $1$, or is there a conceptual explanation? It ...
11
votes
1answer
159 views

Is $e = \sum_n 1/n!$ the most efficient sequence of denominators for rational series for $e$?

The classical series $e = \lim_{n \to \infty} X_n$ where $X_n = \sum_{k=0}^n 1/k!$ is incredibly efficient. But is it known to be the most efficient series in terms of denominators for using fractions ...
11
votes
2answers
770 views

Application de Stone-Weierstrass

Bonjour, J'ai rencontré le problème suivant dans le livre "Real and Functional Analysis" de Lang, au chapitre 3. J'explique d'abord le contexte, puis j'en viendrai à la question précise. Il faut ...
11
votes
1answer
297 views

Approximation of Semicontinuous Functions

Assume that $k \in \mathbb{N}$ and $f : \mathbb{R}^d \rightarrow [0,\infty)$ is lower semicontinuous, i.e. $f(x) \leq \liminf_{y \rightarrow x} f(y)$ for all $x \in \mathbb{R}^d$. Does there exist a ...
10
votes
3answers
913 views

Sine Approximation of Bhaskara

An Indian mathematician, Bhaskara I, gave the following amazing of the sine (I checked the graph and some values, and the approximation is truly impressive.) $$\sin x \approx \frac{{16x\left( {\pi - ...
10
votes
2answers
445 views

How prove this sequence $a_{n}=\sqrt{n}+\frac{1}{2}-\frac{1}{8\sqrt{n}}+o\left(\frac{1}{\sqrt{n}}\right)?$

let sequence $\{a_{n}\}$ such $$a_{1}=1,a_{n+1}=1+\dfrac{n}{a_{n}}$$ show that: $$a_{n}=\sqrt{n}+\dfrac{1}{2}-\dfrac{1}{8\sqrt{n}}+o\left(\dfrac{1}{\sqrt{n}}\right)?$$ This result is china student ...
10
votes
2answers
411 views

Approximation to the Lambert W function

If: $$x = y + \log(y) -a$$ Then the solution for $y$ using the Lambert W function is: $$y(x) = W(e^{a+x})$$ In a paper I'm reading, I saw an approximation to this solution, due to "Borsch and ...
10
votes
5answers
346 views

Why is $e^\pi - \pi$ so close to $20$?

$e^\pi-\pi\approx 19.99909998$ Why is this so close to $20$?
10
votes
5answers
559 views

Approximation theorems

The Weierstrass' approximation theorem for continuous functions on a compact space by using polynomials is well-known. As far as I know, there are some variants of this theorem, e.g. Stone-Weierstrass ...
10
votes
2answers
365 views

Summation of divergent series of Euler: $0!-1!+2!-3!+\cdots$

Consider the series $$\sum\limits_{k=0}^\infty (-1)^kk!x^k\in\mathbb{R}[[x]]$$ Let $s_n(x)$ be partial sum. And let $\omega_{k,n}=(k!^2(n-k)!)^{-1}$. My question is: Prove that ...
10
votes
2answers
265 views

Difference between “≈”, “≃”, and “≅”

In mathematical notation, what are the usage differences between the various approximately-equal signs "≈", "≃", and "≅"? The Unicode standard lists all of them inside the Mathematical Operators ...
10
votes
1answer
399 views

A strange “pattern” in the continued fraction convergents of pi?

From the simple continued fraction of $\pi$, one gets the convergents, $$p_n = \frac{3}{1}, \frac{22}{7}, \frac{333}{106}, \frac{355}{113}, \frac{103993}{33102}, \frac{104348}{33215}, ...
10
votes
3answers
412 views

Solving randomized recurrence relation

I'm looking at the random sequence $x_n$ with $x_0=x_1=1$ and \begin{equation} x_{n+1}=2x_n\pm x_{n-1} \end{equation} where we choose the $\pm$ sign independently with equal probability. Now ...
9
votes
2answers
4k views

Approximating the logarithm of the binomial coefficient

We know that by using Stirling approximation: $\log n! \approx n \log n$ So how to approximate $\log {m \choose n}$?
9
votes
2answers
581 views

Complex Zeros of $z^2e^z-z$

Can anyone give me a hint on showing (in a relatively elegant way, as I know the answer from WolframAlpha), that the complex valued function $z^2e^z-z$ has at most 2 roots with norm less than 2? ...
9
votes
6answers
185 views

When the approximation $\pi\simeq 3.14$ is NOT sufficent

It's common at schools to use $3.14$ as an appropriate approximation of $\pi$. However, here it's stated that for some purposes, $\pi$ should be approximated to $32$ decimal places. I need an example ...
9
votes
1answer
555 views

Tensor products of functions generate dense subspace?

Let $X$ and $Y$ be two spaces in certain category, $F(\cdot)$ a functor associating each space with a function space (with certain topology). Assume that for any $f\in F(X)$ and $g\in F(Y)$, $f\otimes ...
9
votes
1answer
263 views

Why is $10\frac{\exp(\pi)-\log 3}{\log 2}$ almost an integer?

I read that $$10\frac{\exp(\pi)-\log 3}{\log 2} =318.000000033252\dots \approx 318$$ Is this simply a coincidence or can this somehow be explained?
9
votes
1answer
117 views

How could I improve this approximation?

In a computer application, I need to solve trillions of times an equation which can be reduced to $$f(x)=\sin(x)-a x=0$$ Newton methods (quadratic and higher orders) are used for the solution. ...
9
votes
0answers
158 views

Are there known pairs of simple numbers equal to huge precision, but not equal strictly?

Are there known pairs of numbers $a$ and $b$, which at first look at them seemed likely to be equal, and after checking up to $10^n$ decimal places appeared to agree, but suddenly for some $n$ they ...
8
votes
4answers
2k views

Find bound for sum of square roots

Let $a_1,...,a_n$ be real numbers, such that $a_1+...+a_n=A$. What can we say about $\sqrt{a_1}+...+\sqrt{a_n}$? I would like to bound from above thus sum in terms of $A$.
8
votes
6answers
286 views

Approximating $\pi$ with least digits

Do you a digit efficient way to approximate $\pi$? I mean representing many digits of $\pi$ using only a few numeric digits and some sort of equation. Maybe mathematical operations also count as ...
8
votes
4answers
210 views

How could we manually approximate $\sum_{i=1}^{50} i! = 3.1035 \times 10^{64}$?

How could we manually approximate $$\sum_{i=1}^{50} i!$$ to the value $ 3.1035 \times 10^{64}$? I faced this question in my aptitude test,there were four option given,I couldn't solve it during ...
8
votes
3answers
305 views

difference of square roots approximation

In two of my physics courses in the past week, I've come across an approximation for the difference of two square roots for large radicands: $\sqrt{x+a}-\sqrt{x+b}\approx\frac{a-b}{2\sqrt x}$ for ...
8
votes
3answers
449 views

the following inequality is true, but I can't prove it

The inequality $$\sum_{k=1}^{2d}\left(1-\frac{1}{2d+2-k}\right)\frac{d^k}{k!}>e^d\left(1-\frac{1}{d}\right)$$ holds for all integer $d\geq 1$. I use computer to verify it for $d\leq 50$, and find ...