For questions that involve concrete approximations, such as finding an approximate value of a number with some precision. For questions that belong to the mathematical area of Approximation Theory, use (approximation-theory).

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25
votes
7answers
15k 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 ...
125
votes
3answers
7k views

A 1,400 years old approximation to the sine function by Mahabhaskariya of Bhaskara I

The approximation $$\sin(x) \simeq \frac{16 (\pi -x) x}{5 \pi ^2-4 (\pi -x) x}\qquad (0\leq x\leq\pi)$$ was proposed by Mahabhaskariya of Bhaskara I, a seventh-century Indian mathematician. I ...
91
votes
4answers
4k 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 ...
13
votes
2answers
2k 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 ...
73
votes
9answers
5k 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 ...
62
votes
4answers
8k 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 ...
10
votes
1answer
1k 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 ...
30
votes
4answers
1k 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 $?
9
votes
6answers
386 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 ...
6
votes
10answers
1k views

How can I find the value of $\ln( |x|)$ without using the calculator?

I want to know if there is a way to find for example $\ln(2)$, without using the calculator ? Thanks
12
votes
8answers
979 views

Approximation of $e$ using $\pi$ and $\phi$?

$$e \approx \frac{4 \phi +3 \pi-5}{4}$$ where $~\phi~$ is a Golden ratio . Is it possible to construct better approximation of $e$ using $\pi$ , $\phi$ and integers ?
17
votes
5answers
6k 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.
5
votes
3answers
508 views

$\lim_{n\to\infty} f(2^n)$ for some very slowly increasing function $f(n)$

I should be able to answer this myself, but feel insecure anyway. I want to know, whether a function f(n) is bounded if n goes to infinity (and if it's bounded, the limit). Heuristically it appears ...
4
votes
4answers
535 views

Lambert function approximation $W_0$ branch

I am looking for a simple, inexpensive and very accurate approximation of the Lambert function ($W_0$ branch) ($-1/e < x < 0$).
8
votes
4answers
3k views

Why is $22/7$ a better approximation for $\pi$ than $3.14$?

This seems counterintuitive, but $22/7$ is closer to $\pi$ than $3.14=314/100$ which has a significantly greater denominator. Why is $22/7$ a better approximation for $\pi$ than $3.14$? This has ...
9
votes
0answers
468 views

An Expression for $\log\zeta(ns)$ derived from the Limit of the truncated Prime $\zeta$ Function

I think, here, I found $$ P_\color{red}x(\color{blue}s)=\sum_{p<\color{red}x} \frac{1}{p^{\color{blue}s}} =\sum_{\color{green}n=1}^{\infty}\frac{ \mu (\color{green}n)}{\color{green}n} ...
-5
votes
2answers
227 views

Why is $\pi^2$ so close to $10$?

Noam Elkies explained why $\pi^2=9.8696...$ is so close to $10$ using an inequality on Euler's solution to Basel problem $$\frac{\pi^2}{6}=\sum_{k=0}^{\infty} \frac{1}{\left(k+1\right)^2}$$ to form ...
72
votes
14answers
10k 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 ...
24
votes
4answers
1k views

Prove that $2^{2^{\sqrt3}}>10$

With a computer or calculator, it is easy to show that $$ 2^{2^\sqrt{3}} = 10.000478 \ldots > 10. $$ How can we prove that $2^{2^{\sqrt3}}>10$ without a calculator?
17
votes
3answers
2k views

Sine Approximation of Bhaskara

An Indian mathematician, Bhaskara I, gave the following amazing approximation of the sine (I checked the graph and some values, and the approximation is truly impressive.) $$\sin x \approx ...
14
votes
1answer
458 views

Why $e^{\pi}-\pi \approx 20$, and $e^{2\pi}-24 \approx 2^9$?

This was inspired by this post. Let $q = e^{2\pi\,i\tau}$. Then, $$x := \left(\frac{\eta(\tau)}{\eta(2\tau)}\right)^{24} = \frac{1}{q} - 24 + 276q - 2048q^2 + 11202q^3 - 49152q^4+ \cdots\tag1$$ ...
27
votes
9answers
7k 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 ...
20
votes
8answers
5k 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 ...
18
votes
4answers
410 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. ...
10
votes
2answers
1k views

Minimum of the Gamma Function $\Gamma (x)$ for $x>0$. How to find $x_{\min}$?

The $\Gamma (x)$ function has just one minimum for $x>0$ . This result uses some properties of the gamma function: $\Gamma ^{\prime \prime }(x)>0$ and $\Gamma (x)>0$ for all $x>0$ $\Gamma (1)=\Gamma ...
10
votes
3answers
5k views

Approximating the error function erf by analytical functions

The Error function $\mathrm{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\,dt$ shows up in many contexts, but can't be represented using elementary functions. I compared it with another function ...
2
votes
4answers
1k views

Approximating $\pi$ using Monte Carlo integration

I need to estimate $\pi$ using the following integration: $$\int_{0}^{1} \!\sqrt{1-x^2} \ dx$$ using monte carlo Any help would be greatly appreciated, please note that I'm a student trying to ...
6
votes
0answers
226 views

Rational series representation of $e^\pi$

This question is related to Why $e^{\pi}-\pi \approx 20$, and $e^{2\pi}-24 \approx 2^9$? by Tito Piezas III. Andrew Fraker (2014) found an almost-integer which is equivalent to the following ...
4
votes
1answer
199 views

Proof that $\frac{2}{3} < \log(2) < \frac{7}{10}$

Positive integrals $$\int_{0}^{1}\frac{2x(1-x)^2}{1+x^2}dx=\pi-3$$ and $$\int_0^1\frac{x^4(1-x)^4}{1+x^2}dx=\frac{22}{7}-\pi $$ (http://math.stackexchange.com/a/1618454/134791) prove that ...
3
votes
1answer
171 views

$n \approxeq k + 2^{2^k}(k+1)$. How can one get the value of $k(n)$ from this equation?

I am trying to find approximation for this sum. Asymptotic approximation of sum $\sum_{k=0}^{n}\frac{{n\choose k}}{2^{2^k}}$ Doing following way. Let $a_k(n) = \frac{n\choose k}{2^{2^k}}$. I tried to ...
2
votes
1answer
152 views

Mysterious subleading corrections to sums with internal dependence on limit

Is there a standard method for finding expansions in $N$ of sums like $$S(N)=\sum_{n=0}^N \sqrt{N^2-n^2}$$ beyond the first term? It is easy to compute here that $$S(N)=N^2 \int_0 ^1 \sqrt{1-x^2} ...
0
votes
1answer
205 views

Can we prove the Riemann-Lebesgue lemma by using the Weierstrass approximation theorem?

I'd like to prove the following version of the Riemann-Lebesgue lemma: Let $f: [0,1] \to \mathbb R$ be continuous. Then $$\int_0^1 f(x)\sin(nx) \, dx \xrightarrow{n \to \infty} 0$$ It's quite ...
9
votes
0answers
168 views

Could this approximation be made simpler ? Solve $n!=a^n 10^k$

I need to find the smallest value of $n$ such that $$\frac{a^n}{n!}\leq 10^{-k}$$ in which $a$ and $k$ are given (these can be large numbers). I set the problem as : solve for $n$ the equation ...
3
votes
3answers
11k views

Relation between Simpson's Rule, Trapezoid Rule and Midpoint Rule

I am studying numerical approximation and verifying $S_{2n} = \frac{1}{3}\left(T_n +2 M_n\right)$. ($S_n$ refers to Simpson's Rule approximation, $T_n$ refers to Trapezoid Rule approximation and $M_n$ ...
1
vote
0answers
64 views

Series for Stieltjes constants from $\gamma= \sum_{n=1}^\infty \left(\frac{2}{n}-\sum_{j=n(n-1)+1}^{n(n+1)} \frac{1}{j}\right)$

Euler's constant has the following representations (Euler-Mascheroni constant expression, further simplification, http://math.stackexchange.com/a/129808/134791, Question on Macys formula for ...
25
votes
5answers
5k 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? ...
11
votes
2answers
13k views

Explanation and Proof of the fourth order Runge-Kutta method

Runge-Kutte 4th order method is a numerical technique used to solve ordinary differential equation of the form $dy/dx=f(x,y), y(0)=y_0$ It gives $y_{i+1}$ in the form $y_{i+1} = ...
35
votes
1answer
496 views

Is there an integral for $\pi^4-\frac{2143}{22}$?

In Ramanujan's Notebooks, Vol 4, p.48 (and a related one in Quarterly Journal of Mathematics, XLV, 1914) there are various approximations, including the close (by just $10^{-7}$), $$\pi^4 \approx ...
13
votes
2answers
2k views

Ramanujan's approximation for $\pi$

In 1910, Srinivasa Ramanujan found several rapidly converging infinite series of $\pi$, such as $$ \frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum^\infty_{k=0} \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}. ...
4
votes
1answer
682 views

On finding polynomials that approximate a function and its derivative (extensions of Stone-Weierstrass?)

The Stone-Weierstrass Theorem tells us that we can approximate any continuous $f:\mathbb{R}^n\to\mathbb{R}$ arbitrary well on a compact subset of $\mathbb{R}^n$ by some polynomial. Suppose that $f$ is ...
4
votes
1answer
2k views

Method for estimating the $n^{th}$ derivative?

When using numerical analysis, I often find that I am required to estimate a derivative (e.g. when using Newton Iteration for finding roots). To estimate the first derivative of a function $f(x)$ at ...
15
votes
2answers
461 views

Intuitively, why is the Euler-Mascheroni constant near sqrt(1/3)?

Questions that ask for "intuitive" reasons are admittedly subjective, but I suspect some people will find this interesting. Some time ago, I was struck by the coincidence that the Euler-Mascheroni ...
8
votes
5answers
6k views

Approximating Logs and Antilogs by hand

I have read through questions like Calculate logarithms by hand and and a section of the Feynman Lecture series which talks about calculation of logarithms. I have recognized neither of them useful ...
6
votes
1answer
108 views

Is there a series to show $22\pi^4>2143\,$?

This extends this post. I. For $\pi^3$: $$\pi^6-31^2 =\sum_{k=0}^\infty\left(-\frac{63}{(2k+2)^6}+\frac{31^2}{(2k+3)^6}\right) =\sum_{k=0}^\infty P_1(k)\tag1$$ As pointed out by J. Lafont, when ...
1
vote
6answers
4k views

How to calculate square root or cube root? [duplicate]

I was reading Richard Feynman biography when I read that one time he was able to calculate the cube root of large number in his brain by just using simple facts of everyday life. So my question is ...
13
votes
1answer
261 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 ...
6
votes
2answers
3k views

How can I calculate non-integer exponents?

I can calculate the result of $x^y$ provided that $y \in\mathbb{N}, x \neq 0$ using a simple recursive function: $$ f(x,y) = \begin {cases} 1 & y = 0 \\ (x)f(x, y-1) & y > 0 \end ...
4
votes
3answers
138 views

Why do we need an integral to prove that $\frac{22}{7} > \pi$?

We know this famous (and beautiful) integral which shows that $\dfrac{22}{7} > \pi$ as : $$0 < \int_0^1 \frac{x^4(1-x)^4}{1+x^2} \, dx = \frac{22}{7} - \pi$$ Now since the integrand is ...
18
votes
3answers
7k 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 $\log {m \choose n}$?
6
votes
1answer
2k views

Approximating $\arctan x$ for large $|x|$

I would like to know if there is reasonably fast converging method for computing large arguments of arctan. Until now I've came across Taylor series that converges only on interval $(-1,1)$ and for ...