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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4
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0answers
96 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 $\pi^3 \gt 31$ There are series with all terms positive for $\pi-3$ and $10-\pi^2$ ...
0
votes
0answers
10 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
19 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 ...
-5
votes
2answers
185 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 ...
5
votes
1answer
169 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 ...
0
votes
0answers
11 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
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 ...
0
votes
1answer
50 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
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0answers
15 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 ...
3
votes
0answers
40 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: ...
2
votes
1answer
151 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 ...
0
votes
0answers
27 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
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
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
2answers
27 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 $N$; ...
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 ...
1
vote
1answer
72 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 ...
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 ...
3
votes
1answer
116 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
0answers
38 views

Sum Calculation: $\sum_{n=1}^\infty \left(1- \frac{\cosh^{-1} n}{\log 2x}\right)$

I was investigating the asymptotic properties of the $\cosh$ functions and how they all strongly relate to $e^x$ In my studying, I found out that $\cosh x\sim \frac{e^x}{2}$ By that definition, that ...
0
votes
0answers
31 views

What approximations for the Gamma function's inverse appear to work 'best'?

So I was wondering how we approximate the inverse of the Gamma function, where I tried a few methods: Lagrange inversion theorem: $$\Gamma^{-1}(z)=a+\sum_{n=1}^{\infty}\lim_{w\to ...
0
votes
1answer
231 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 ...
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 ...
1
vote
2answers
39 views

Error when approximating $\int_0^{1}(x^{2}+x)dx$ with midpoint rule.

Task is to define the exact error when approximating $\int_0^{1}(x^{2}+x)dx$ with midpoint rule using n subintervals. I know the error term is $E(f)=\frac{1}{24}(b-a)f^{''}(\varepsilon)h^{2}$ but im ...
0
votes
2answers
50 views

Bound for $\log { \binom{n}{i}}$?

(1) Are there better (smaller; tighter) bounds for $\log { \binom{n}{i}}$, than $O(n \log n )$? (2) Under what conditions $O(i \log n)$ is a good bound? Clearly this bound should be in a way that it ...
0
votes
1answer
39 views

Using Lagrange Remainder to find the approximation of $\sqrt(8)

I'm looking for an approximation of $\sqrt 8$ with an approximation of $10^{-4}$. It was given that $\sqrt 8 =3\sqrt\frac{8}{9}$ so I set up a general series for $ \sqrt{1+x}^\frac{1}{2} $ around zero ...
0
votes
0answers
16 views

What do you call Binomial Distributions Correlated like this?

If we have two biased coins, and throw both $n$ times, then we can describe the probability of getting $a$ heads with the first coin and $b$ heads with the second coin, as a product distribution of ...
0
votes
1answer
24 views

Use linear scales for both the X- and Y- axes?

I am trying to understand this question: Use linear scales for both the X- and Y- axes. In which region, ƒ2(x) is a good approximation to ƒ1(x)? My equations are: ...
0
votes
0answers
25 views

Approximating a Riccati Differential Equation/Differential equation of two variables

I am currently really struggling to solve/approximate this Riccati equation $x'(t, v) = cv(1-x(t)) + \beta x(t)(1-x(t)) - \gamma x(t)$ st. $0 < c, \gamma, x_{0} < 1$ $ 0 \leq v \leq 1$ $ 0 ...
2
votes
2answers
71 views

Closed form or approximation of $\sum\limits_{i=0}^{n-1}\sum\limits_{j=i + 1}^{n-1} \frac{i + j + 2}{(i + 1)(j+1)} (i + 2x)(j +2x)$

During the solution of my programming problem I ended up with the following double sum: $$\sum_{i=0}^{n-1}\sum_{j=i + 1}^{n-1} \frac{i + j + 2}{(i + 1)(j+1)}\cdot (i + 2x)(j +2x)$$ where $x$ is some ...
1
vote
0answers
48 views

Three almost-integers of the form $ce^{H_a+H_b}\approx 2^k\pm1$

The approximation $$H_n\approx log(2n+1)$$ http://math.stackexchange.com/a/1602945/134791 suggests that the harmonic number for composite odd numbers might be close to the sum of the harmonic ...
2
votes
0answers
56 views

Integral of a Gaussian with Trigonometric functions Involved

I am having a difficult time evaluating an integral unlike any integral I have seen before. To get right into things here is the integral: $$\frac{A}{\sigma_o\sqrt{2\pi}}\int_{-\infty}^\infty ...
1
vote
0answers
13 views

Cubic curve as approximation of Euler spirals?

I was reading the wiki article about Euler spirals and I reached this passage: Unaware of the solution of the geometry by Leonhard Euler, Rankine cited the cubic curve (a polynomial curve of ...
2
votes
1answer
30 views

How many “seeks” are there in these binary sequences?

Consider a set of $k \geq 1$ random, IID binary sequences of length $n$, denoted $S_i,\;i = 1\ldots k$, and a "master sequence", also of length $n$, and denoted $S_M$ (see figure for $k = 4$).   ...
1
vote
1answer
80 views

Can this integral be evaluated/approximated?

I've been trying to evaluate this integral without much success: $\displaystyle \int_{-\infty}^\infty dx\, e^{iax} \frac{1- e^{-c\sinh^2 bx}}{\sinh^2 bx}$ I've tried contour integration. There are no ...
0
votes
0answers
97 views

How to verfy if the approximations of the complex error function have no poles?

I found an article published few days ago in arXiv:1601.01261 that shows a very simple Matlab code for computation of the complex error function (aka the Faddeeva function) defined as \begin{equation} ...
0
votes
0answers
15 views

Galerkin and Ritz method

I know Galerkin method gives us the best approximation. But i couldn't understand the difference between this method and Ritz method for approximation. Do not give us the same results? What is the ...
1
vote
0answers
63 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 ...
2
votes
0answers
183 views

Why is $e$ close to $H_8$, closer to $H_8\left(1+\frac{1}{80^2}\right)$ and even closer to $\gamma+log\left(\frac{17}{2}\right) +\frac{1}{10^3}$?

The eighth harmonic number happens to be close to $e$. $$e\approx2.71(8)$$ $$H_8=\sum_{k=1}^8 \frac{1}{k}=\frac{761}{280}\approx2.71(7)$$ This leads to the almost-integer ...
0
votes
0answers
13 views

looking for approximation/expansion of $f(t)=a(t)/\sqrt{b(t) + \epsilon(t)}$ with $\epsilon(t) << b(t)$

I have the following function $ f(t) = \frac{a(t)}{\sqrt{b(t) + \epsilon(t)}} $ defined for $t\geq 0$. I know that $a(t) > 0$, $b(t) > 0$, $\epsilon(t) \geq 0$ and $\epsilon(t) << b(t) $ ...
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 ...
6
votes
0answers
225 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 ...
2
votes
2answers
34 views

Approximation of $2$nd Derivative Up to $O(h^4)$

Investigate if it is possible to obtain 4th order accuracy using 5 points for a 2nd derivative approximation, i.e. is it possible to determine a, b, c, d, e in $$y''(0) = ...
1
vote
0answers
23 views

Aproximation (Proof) [duplicate]

Can you help me? If $f$ and $f'$ are continuous at $[a,b]$ (where $a,b\mathbb{\in R}$), then $\forall\epsilon>0$ exists a polynomial $p$ such that $\left\Vert f-p\right\Vert _{\infty}\leq\epsilon$ ...
0
votes
0answers
46 views

integral involving error function (erf)

Does anybody know if a closed form of this integral exist? $\int \mbox{erf}(x) \ln(\mbox{erf}(x)) \Bbb dx$ where erf is so called error function. In case there is no closed form solution. Is it ...
1
vote
0answers
55 views

Why is $\sum{\frac{1}{(j/2)!}(\xi)^j}\approx\sum{\frac{1}{j!}(\xi)^{2j}}$?

Why is $\sum{\frac{1}{(j/2)!}(\xi)^j}\approx\sum{\frac{1}{j!}(\xi)^{2j}}$? This equation is used for solving the Schrödinger equation of the harmonic oscillator at one point in Griffiths, ...
0
votes
2answers
49 views

Prove that if $2a^2 - b^2 = \pm1$ then $\frac ba \approx \sqrt2$ [closed]

Prove that if $2a^2 - b^2 = \pm1$ then $\frac ba\approx\sqrt2 $ (a,b) (1,1),(2,3),(5,7),(12,17),(29,41),(70,99)....
1
vote
2answers
45 views

How to estimate ln(1.1) using quadratic approximation?

So the general idea for quadratic approximation is assuming there a function $Q(x)$ we want to estimate near $a$: $Q_a(a) = f(a)$ $Q_a'(a) = f '(a)$ $Q_a''(a) = f ''(a)$ But then how do you ...