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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10
votes
2answers
210 views

Is $\pi^k$ any closer to $\mbox{nint}(\pi^k)$ than expected?

Particular questions such as Why is $\pi$ so close to $3$? or Why is $\pi^2$ so close to $10$? may be regarded as the first two cases of the question sequence Why is $\pi^k$ so close to ...
0
votes
0answers
26 views

approximation of a trigonometric sum

I have a trigonometric sum as below $$\sum_{r=0}^{N-1}\frac{\sin^2(\frac{\pi(Ne-e-r+n))}{N})}{\sin^2(\frac{\pi(r-n+e))}{N})}$$ and I want to show analytically that for small $e$ ($e<0.2$) and large ...
1
vote
1answer
106 views

A series to prove $\frac{22}{7}-\pi>0$

After T. Piezas answered Is there a series to show $22\pi^4>2143\,$? a natural question is Is there a series that proves $\frac{22}{7}-\pi>0$? One such series may be found combining ...
0
votes
2answers
44 views

About Taylor series

Suppose $f(0) = 0, f'(0) = 2, f''(0) = −1$ and $|f''' (x)| ≤ 0.024$ for $0 ≤ x ≤ 2$. Estimate $f(1)$ to $4$ significant figures by using a Taylor polynomial. Compute a good bound for the absolute ...
6
votes
1answer
110 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
1answer
45 views

Another way to calculate $\cos(x)$ and $\cosh (x)$

I usually don't see any expressions for approximate calculation of trigonometric functions aside from their Taylor series. But Taylor series work better for small angles, so in general they are not ...
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 ...
1
vote
0answers
36 views

Best Approximate Solution of Heat Equation (Diffusive Logistic Equation)

If $u(x,t) \ge 0$ in the domain $ (0 \times 1) \times (0,\infty)$, find a function that caps the value of $u(x,t)$ in the region $(0 \times 1) \times (0,T)$. $u(x,t)$ is a solution of the following ...
4
votes
2answers
218 views

Is there an alternative to Taylor expansion of functions with more control over the error distribution?

As a physics student I see the Taylor series being (ab)used very often, mostly for the expansion of $\exp(x)$. The usual line of though is that the error (remainder) term of a high order is likely ...
1
vote
0answers
26 views

Approximating $\prod_{r=s}^t (1-b/r)$

I am currently trying to place an order of precision on the approximation $$\prod_{r=s}^t \left(1-\frac{b}{r}\right) \approx \left(\frac{s}{t}\right)^b$$ This follows because $$\prod_{r=s}^t ...
0
votes
2answers
86 views

Find to how many digits the value $\frac{355}{113}$ is an accurate approximation of $3.1415929204$.

Find to how many digits the value $\frac{355}{113}$ is an accurate approximation of $3.1415929204$. What i did was i computed it using a calculator and got the of $\frac{355}{113}$ to be ...
4
votes
1answer
200 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 ...
5
votes
0answers
174 views

A monotonically increasing series for $\pi^6-961$ to prove $\pi^3>31$

This question is motivated by Why is $\pi$ so close to $3$?, Why is $\pi^2$ so close to $10$? and Proving $\pi^3 \gt 31$. I. $\pi$ and $\pi^2$ There are series with all terms positive for ...
0
votes
0answers
27 views

Searching an Approximation Formula for Two Parameters

I have an algorithm with two parameters ($p_1$ and $p_2$) and one result ($x$). Interesting (for me) parameters and results are: ...
-5
votes
2answers
229 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 ...
1
vote
2answers
50 views

How to find approximate value of $1.01e^{1.01({0.99) }^2} $?

I want to find the approximate value of $1.01e^{1.01({0.99) }^2}$ by using derivative. I tried choosing x=1 and $\delta x=0.01$ it didnt work. How can I start?
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 ...
0
votes
0answers
13 views

Continuity of Monte-Carlo simulations with uniformly distributed input parameters

Suppose a continuous and monotone function $f:\mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$ to be given. So, in the general case, if I slightly change parameters $a$ and $b$, the function ...
3
votes
2answers
20 views

Finding a root approach with a polynomial

So, i'm solving last's year's exams in Mathematical Analysis and i've found one interesting. It says: The equation $e^{-4x}=5x^2$ has one root close to (nearby) 0. By approaching $e^{-4x}$(close to ...
0
votes
0answers
30 views

Find required degree of Maclaurin polynomial to estimate the cosine to two decimal places

I have a question where I am asked to find the amount of terms required in a Maclaurin polynomial to estimate $\cos(1)$ to be correct to two decimal places. So far what I have done is used Taylor's ...
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 ...
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, $$x := \left(\frac{\eta(\tau)}{\eta(2\tau)}\right)^{24} = \frac{1}{q} - 24 + 276q - 2048q^2 + 11202q^3 - 49152q^4+ \cdots\tag1$$ ...
0
votes
1answer
239 views

Is there an integral or series for $\frac{\pi}{3}-1-\frac{1}{15\sqrt{2}}$?

The approximation $$\pi\approx\frac{22}{7}=3+\frac{1}{7}$$ suggests that the closest integer to $\frac{1}{\left(\pi-3\right)}$ is $7$. However, $$ \frac{1}{\left(\pi-3\right)^2}\approx49.879 $$ is ...
0
votes
0answers
12 views

approximating random variable

Can anyone explain the construction of the sequence of simple random variables that can be approximated to any random variable? $ X_n(\omega)=\sum_{k=0}^{n2^{n}} k2^{-n}$ where $\, k2^{-n} \leq ...
1
vote
1answer
44 views

Maclaurin series of $e^{-x^2}$ Error

The task is to first estimate the second degree Maclaurin series of $e^{-x^2}$ and thus estimate the integral of the function from $0$ to $0.5$. This part is no problem. The following task is to ...
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 ...
1
vote
0answers
32 views

Mixing Fuzzy Logic and Probabilistic interpretation of a dataset

A probabilistic data cloud is a set $M$ of data points $\{m_i\}_i$, where each data point $m_i$ is associated to an event $E_i$ expressing the set of the occurrences of $m_i$ in any possible ...
1
vote
1answer
260 views

Numerical solution of non-linear differential equation with MATLAB

I need some information to know if I can solve a nonlinear integral equation with terms $ u_{x} $ , $ u_{x}.u_{y} $ , $ u_{xx} $ , $ u_{xy} $ $u_{yy} $ $ u_{x}^{2} $ $ u_{y} ^{2} $ By numerical ...
2
votes
1answer
161 views

A series related to $\pi\approx 2\sqrt{1+\sqrt{2}}$

This question follows a suggestion by Tito Piezas in Is there an integral or series for $\frac{\pi}{3}-1-\frac{1}{15\sqrt{2}}$? Q: Is there a series by Ramanujan that justifies the approximation ...
4
votes
0answers
43 views

How can I use Stirlings inequality to prove this inequality?

Let $p,k$ be natural numbers with $p\ge k$, show that $$ \frac{(p-k)!}{(p+k)!} \le \frac{1}{p^{2k}} \left(\frac{e}{2} \right)^{2k^2/p}. $$ The text where I come across this says to use Stirling's ...
0
votes
1answer
51 views

small amplitude oscillation of rotating system.

I've solved the euler-lagrange equation for a frictionless bead on circular vertical loop of radius a where the loop is rotating at $\Omega$ to get the equation of motion for the bead as ...
0
votes
0answers
29 views

Determining the minimal number of terms to use in a sum to approximate a number given a tolerance

In page 33-34 of Numerical Analysis by Burden & Faires an algorithm was given to compute the minimal value of $N$ for which $$|\ln{1.5}-P_N(1.5)|<10^{-5}\tag{1}$$,where ...
0
votes
0answers
16 views

functional analysis (Faedo Galerkin Method)

if \begin{equation} \left\{ \begin{array}{l} (u^{0\nu },u^{1\nu },v^{0\nu },v^{1\nu },p^{0\nu },q^{0\nu }) \rightarrow (u^{0},u^{1},v^{0},v^{1},p^{0},q^{0}) \\ \text{strongly in } (H^1_\gamma \cap ...
4
votes
0answers
46 views

Is there a section of mathematics that studies near-integer equations.

When I saw: $$e^\pi-\pi \approx 20$$ I thought it was pretty cool. And : $$\pi^3 \approx 31$$ So now the thought comes to me is what positive integer value of $n$ will make the expression: ...
13
votes
1answer
578 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 ...
19
votes
2answers
646 views

Approximation for $\pi$

I just stumbled upon $$ \pi \approx \sqrt{ \frac{9}{5} } + \frac{9}{5} = 3.141640786 $$ which is $\delta = 0.0000481330$ different from $\pi$. Although this is a rather crude approximation I ...
3
votes
1answer
117 views

Calculating ${(0.9)}^{\left(0.6\right)}$ with an approximation of ${10}^{\left(-4\right)}$

I'm having extreme difficulties understanding how to use Lagrange theorem to find an approximation. So far for my series I have: $$(1+(-x))^\frac{3}{5}= ...
0
votes
1answer
29 views

Is $e^{-r/2}$ equivalent to $r^{-(l+1)}$ in the radial solution of Laplace equation?

When we solve the Laplace equation for Hydrogen Wave Equation at large r, we obtain the expression below to account for the behavior of the wave at very very large $r$ $$R=e^{-(r/2)}$$ At very small ...
0
votes
1answer
18 views

Which approximation should I use?

I have a function $ k(x,y)$, and I want to approximate it for low values of x and y. $k(x,y) = \dfrac{a^3-ax^2-x^3+a^2x+ay^2-xy^2}{a^3-ax^2+x^3-a^2x+ay^2+xy^2}$ With $ a>>x, a>>y $ ...
0
votes
2answers
28 views

Approximating a sum of reciprocals

What is a good approximation for the function: $$S_{N,k} = \sum_{i=N}^\infty {\frac{1}{i^k}}$$ when $k$ is a given constant (2, 3 or 4) and $N$ is large? $S_{N,k}$ is a decreasing function of $N$; ...
4
votes
1answer
35 views

QR(pivot) vs SVD for low rank approximation

Define the low rank problem as finding the approximation of matrix A, B: where we want to minimize rank(B) and we want the 2 norm of the residu of A-B to be less than epsilon. Could someone help me ...
0
votes
1answer
36 views

When is right to kill $r^l$ and/or $r^{(-l-1)}$?

When we solve the Laplace equation in spherical polar coordinate, we get the radial part whose solution is: $$R=Ar^l+Br^{-(l+1)}$$ Now, some solutions keep this two terms, but when we derive the ...
0
votes
5answers
793 views

linear or quadratic approximation for exp(-x) for large x

Is there any linear or quadratic approximation of $exp(-x)$ where $0<x<L$ ? $L$ is large, may be 40 (say) i.e. $x$ is not close to zero.
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 ...
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 ?
1
vote
1answer
74 views

Simpler derivation to $\pi$ [closed]

I'm an amateur in mathematics, being in 9th grade. I have been trying to derive $\pi$. During this I reached a limit to find the value of $\pi$. $$\lim_{x \to 0} \frac{180\sin x}{x}$$ Where $x$ is in ...
1
vote
1answer
30 views

An approximation for the Lambert W-function

Proposition Let $f(x) = k^{x}x$, where the values of both $f(x)$ and $k$ are known. Let $x_{0} = f(x)$, and: $$x_{n + 1} = \frac{1}{2}\log_{k}{\left(\frac{k^{x_{n}}x_{0}}{x_{n}}\right)}$$ ...
0
votes
0answers
14 views

Continuous/interpolating alternative to order of magnitude?

Define $\operatorname{magnitude}\left(x\right) = 10 ^ { \lfloor \log_{10} x \rfloor }$ and $\operatorname{magnitude'}\left(x\right) = 10 ^ { \lfloor \log_{10} x \rceil }$ Currently I'm using this ...
1
vote
1answer
83 views

Which function to kill: Sine or Cos?

I got an equation which was a solution to some familiar Differential Equation I am solving, the solution takes the form of: $$V=Ce^{-ix}$$ but $$Ce^{-ix}=A\cos(x)+B\sin(x)$$ so ...
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 ...