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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50 views

Obtaining exact decimals in bisection method

While studying the bisection for the approximation of roots of non-linear equations I was given the following bound for the error: $|x_n-s| \leq \frac{(b-a)}{2^{n+1}}$ where $x_n$ is the n-th ...
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3answers
46 views

Approximated second derivative

Approximated second derivative of $y(0)''$ function $y(x)$ at x = 0 difference quotient, using the values ​​of $y(x)$ at the sites of the three-point template $ {x}_{1} = \frac {-4h} { 5}, {x} _ {2} = ...
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0answers
21 views

Approximated second derivative

Approximated second derivative of $y(0)''$ function $y(x)$ at x = 0 difference quotient, using the values ​​of $y(x)$ at the sites of the three-point template $ {x}_{1} = \frac {-4h} { 5}, {x} _ {2} = ...
1
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0answers
29 views

Unbounded approximation algorithm for minimum vertex cover

Suppose we find the minimum vertex cover of a graph by repeatedly choosing the vertex with the highest degree and delete all edges incident on that vertex, until there are no edges left. How can one ...
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0answers
71 views

Infinite products of even analytic functions - highly accurate approximation

I discovered a way to evaluate infinite products of even analytic functions with high accuracy. $$ \prod_{k=1}^{\infty} f(k^2) \approx \prod_{k=1}^{\infty} \left(1-\frac{A_1}{k^2}+\frac{A_2}{k^4}-\...
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0answers
21 views

Step in using Stirling's formula to get an upper bound

I'm having trouble seeing why the following holds. Given the conditions that $N=\binom{n}{2}$ and $m$ is a function of $n$ such that $N-m \to \infty$ as $n \to \infty$, why is it that $$(1+o(1)) \...
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1answer
23 views

Reverse economization of Chebyshev series

Suppose I have some function which is represented as converging series of Chebyshev polynomials of first kind in $[-1;1]$: $$ f(x)=\sum\limits_{n=1}^\infty a_n T_{2n}(x) $$ I need to transform this ...
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1answer
34 views

Density of polynomials in $\cos t$ in $\mathcal{C}^0([0,\pi],\mathbb{R})$

I'm looking at the Fourier cosine transform, and as a preliminary I have to show that every $f$ in $\mathcal{C}^0([0,\pi],\mathbb R)$ is the uniform limit of a sequence of functions of the form $t\to ...
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1answer
74 views

Why is $\tan 3 + \pi$ a near-integer? [closed]

When playing with my calculator I found that $$\tan 3 + \pi \approx 3$$ Is there a mathematical reason for this?
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3answers
178 views

Something similar to the bizarre Koide formula?

In 1981, Koide found the empirical relation, $$\frac{m_e+m_\mu+m_\tau}{\big(\sqrt{m_e}+\sqrt{m_\mu}+\sqrt{m_\tau}\big)^2} = 0.666659\dots\approx \frac{2}{3}\tag1$$ where $m$ are the masses of the ...
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0answers
27 views

Find the linearization of the function at 0

The problem asks: Find the linearization of $f(x)= \sqrt{a+bx} $ at $0$ To get all parts of $L(x) \approx f(c) - f'(c)(x-c)$ I've done: $$f(0) = \sqrt{a}$$ $$f'(0) = {b\over 2\sqrt{a}} $$ Now: $$...
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1answer
44 views

Find alternative shortest paths given extra properties

This is a follow-up question for a question I asked at here. The problem is mapped to a graph with say non-negative weights on edges (no preference if it can be directed or not). However, along with a ...
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0answers
28 views

Interpolating sequence problem.

Below is a question which I cannot quite figure out. Any tips would help appreciated!I've been working on this for about a month. "Let $(z_j)$ be a sequence of distinct points in a domain $D$ that ...
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1answer
19 views

Taylor expansion of this electric field

I'm trying to determine what happens when R>>z for the below equation $\frac{z\sigma}{2\varepsilon }\left ( \frac{1}{z}-\frac{1}{\sqrt{R^{2}+z^{2}}} \right )$ Like most books, which is a great ...
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1answer
13 views

Minimal error given when making an approximation of $f(x)$ by sines and cosines

I am studying by myself Fourier analysis and have encountered the following problem: We are trying to approximate a function by a finite sum of sines and cosines with general constant coeficients: $$...
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3answers
71 views

Approximating $x=\sqrt{2}+1$

Suppose $y>1$ is some approximation to $x=\sqrt{2}+1$. Give a brief reason (not a proof) why one should expect $(1/y)+2$ to be a closer approximation to $x$ than $y$ is. After testing this out ...
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1answer
26 views

Show a bound on the error

Suppose $g \in C[a,b]$, and $M(b-a) < 1$, and let $y$, $y_N$ $\in C[a,b]$ be the unique solutions of the equations, $y = g + Ky$ and $y_N = g + K_N y_N$. $y_N$ has been previously defined as an ...
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3answers
48 views

A simple approximation algorithm?

I'm not sure if this method works perfect, but I have found it to work in approximating things easily, that is, you need no more than simple algebra to understand this method. Suppose you are trying ...
1
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0answers
42 views

How can I prove two empirically derived graphs are topologically equivalent?

I have two graphs that I've derived from an empirical data set and I suspect that they're topologically equivalent. It seems much easier to show that these graphs are not equivalent than to show that ...
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0answers
39 views

approximation of a trigonometric sum

I have a trigonometric sum as below $$\frac{1}{N^2}\sum_{r=0}^{N-1}\frac{\sin^2(\pi e)}{\sin^2(\frac{\pi(r-n+e))}{N})}\frac{\cos^2(\frac{\pi(Ne-e-r+n)}{N})}{\cos^2(\frac{\pi(Ne-e)}{N})}$$ and I want ...
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3answers
601 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 ...
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2answers
47 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 ...
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2answers
430 views

Is $\pi^k$ any closer to its nearest integer 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 its nearest ...
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1answer
72 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 ...
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1answer
192 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 ...
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0answers
29 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 \left(1-\...
2
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1answer
75 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 ...
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1answer
49 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: ...
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2answers
127 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?
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0answers
232 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 $\...
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0answers
20 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 $f(a,...
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2answers
23 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)...
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0answers
37 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 ...
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2answers
320 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 ...
4
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1answer
239 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<\pi&...
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0answers
14 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 X(\...
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0answers
40 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 non-...
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1answer
64 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 $$\frac{d^{2}\...
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0answers
42 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 H^...
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1answer
98 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: $$\pi^...
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1answer
196 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 $\pi\...
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1answer
21 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 $ ...
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1answer
30 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 ...
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2answers
31 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 $N$; ...
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1answer
37 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 ...
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1answer
81 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 ...
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0answers
18 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
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1answer
84 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 $$V=A\cos(x)+B\sin(...
3
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1answer
121 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}= 1-\frac{3}{5}x-\frac{3}{25}x^2-\frac{7}{125}x^...
0
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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 ...