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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5
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2answers
56 views

Find sum of series $\sum _{ n=1 }^{ \infty }{ { (-1) }^{ n+1 }\left( \frac { n+2 }{ { n }^{ 5 } } \right) } $ correct to 3 decimal places.

This is a question I came across while studying on Khan Academy: Find the sum of the series $\sum _{ n=1 }^{ \infty }{ { (-1) }^{ n+1 }\left( \frac { n+2 }{ { n }^{ 5 } } \right) } $ correct to ...
0
votes
0answers
72 views

Fourier series convergence and error of approximation

Given $f(t) = e^{-t}$, $|t|\leπ$, Determine the the $3rd$−order Fourier series and then calculate to what values does the Fourier series for the $2π$-periodic function $f(t)$ converge in $|t| ≤ π$ ...
1
vote
0answers
48 views

If $0.2349$ is approximated to $0.2299$, what is number of significant figures in such approximation?

If $0.2349$ is approximated to $0.2299$, what is number of significant figures in such approximation? Let $x_T=0.2349$, $x_A=0.2299$, then absolute error = $|x_T-x_A|=0.0050=\frac{1}{2}\times ...
1
vote
1answer
53 views

Bootstrap approximation sample mean: skip the centering?

I know different forms of this question has been asked before on this website, but I don't know how to apply the answers there to this specific form. Anyway, let $X_1, ..., X_n \sim F$ be iid, with ...
0
votes
1answer
42 views

Trouble plotting Fourier Series in MATLAB

I was wondering if anybody could help me with plotting my Fourier Series in MATLAB. I've had a go at it and I don't believe I have arrived at the correct answer. I've plotted the expanded result fine ...
-3
votes
2answers
72 views

Find an approximate value of the sine of 61 degrees

Use the fact that $\sin 61° = \sin (60°+1°)$ to give an approximate value of $\sin 61°$ in terms of $\pi$. My textbook says that the answer is $(180\sqrt(3)+\pi)/360$, but I don't understand how ...
0
votes
3answers
89 views

What is the function for a 'fractal sine wave'?

Maybe I abused the word fractal here. I was wondering what's the function ( if not functions ) for this wave: My attempt was this function, It looks the same, but It's not. The second sine wave is ...
1
vote
2answers
55 views

Numerically stable evaluation of $x\ln(x)$

I have numerical difficulties with the function $$x\ln(x/x_0)-x+x_0$$ with $x\ge0$ and $x_0>0$ since it bears the evil evaluation of $\ln(x)$ for $x\rightarrow0$ which is the domain of interest for ...
1
vote
0answers
35 views

Least-squares fit of a nonlinear (polar) system

I want to determine the six unknown coefficients (uppercase letters) of the model $$x=X_c+(Au+B)\cos(Cv+D),\\y=Y_c+(Au+B)\sin(Cv+D)$$ given a set of data $(x_k,y_k,u_k,v_k)$, by least-squares. As ...
1
vote
0answers
23 views

How is inequality/approximation obtained?

I am reading on combinatorics - probabilistic methods. In one particular problem I came across the inequality $$\binom{n}{k}(1-2^{-k})^{n-k} < n^k e^{-(n-k)2^{-k}}$$ I understand that ...
0
votes
1answer
38 views

Write $a$ as a function of $n$ when $\sum_{i=1}^{n} (i + a)^{-1} = 1$

Is there a good integral estimation technique I can apply here? Thanks!
1
vote
1answer
13 views

Algebraic Error In My Work for Secant Method

I keep getting the wrong answer for problems involving approximating roots, mostly due to some kind of algebraic oversight I am making. Essentially, given a problem where I need to approximate a value ...
1
vote
2answers
53 views

find an approximate solution, up to the order of epsilon

The question is to find an approximate solution, up to the order of epsilon of following problem. $$y'' + y+\epsilon y^3 = 0$$ $$y(0) = a$$ $$y'(0) = 0$$ I tried to solve the given problem using ...
0
votes
0answers
54 views

Find the approximations T4 and M4 and give error bounds.

a.Find the approximations T4 and M4 for Integral from 1 to 2 35e^(1/x) b. Estimate the errors in the approximations of part (a). (Round your answers to six decimal places.) For this park you use the ...
3
votes
1answer
40 views

Rigorous rationale for the Pade Approximant?

I recently asked a Question for which the only Answer I got was a recommendation to punt and use the Pade Approximant. This is the first time I recalling seeing this, and intuitively it seems like ...
0
votes
0answers
11 views

Question on Continuity Correction for Hypothesis Testing

This is a question on continuity correction in hypothesis testing for the approximation of Binomial to Normal Distribution: (i) Deng wishes to test whether a certain coin is biased so that it is ...
0
votes
0answers
35 views

Is this a reasonable approximation

I have quarterly return data, $r$ and am looking to approximate the mean and variance of annual returns, $R$. They are linked by the following formula: $$R=(1+r)^4-1$$ I am wondering if the ...
1
vote
0answers
21 views

Piecewise linear approximation of a (low order) trigonometric polynomial, with quantization

The Problem Given a trigonometric polynomial of order $K$: $$y(t)=\sum_{k=-K}^K c_k \ e ^ {j k \bar \omega t} \ , \qquad c_{-k}=\bar c_k$$ we want to find the best approximation to it using a ...
0
votes
0answers
35 views

Why are the disadvantages of approximation?

When I do 1/3 = 0.33333 but when I do 3*0.33333 then answer is 0.99999, I mean not whole 1 but 0.1 less than 1. What are the drawbacks of this think/rule since it's very basic math. Also why One cant ...
0
votes
2answers
62 views

How to approximate the division by a number like prime number?

I was solving some mathematical questions and have come across the situation, where I need to divide 3900/139. Here is my question, a. Can I assume 139 to 140 for the ease of division? If so, how ...
5
votes
1answer
86 views

Verify proof of $ \left( 1 + \frac{x}{n} \right)^n < e^x \,\text{and} \, e^x < \left(1 - \frac{x}{n}\right)^{-n}$

The following is from Tom Apostol's Calculus I, on page 250, exercise 42.: If $\mathit{n}$ is a positive integer and if $\mathit{x} > 0$, show that $$ \left( 1 + \frac{x}{n} \right)^n < e^x ...
0
votes
3answers
83 views

Determining the parameters of a differential equation

Let's assume: $$ay''(t)+by'(t)+cy(t)=f(t)$$ And we are given $n$ points $\{(t_1,y_1),(t_2,y_2),\ldots,(t_n,y_n)\}$, where $y_i=y(t_i)+\epsilon_i$. By approximation of the differential equation, I ...
2
votes
3answers
78 views

Approximation of the solution $n$ to the equation $n \log n = 100\,000$ [closed]

How can I find an approximation value for the value of $n$ for which $$n \log(n)=100\,000$$ or each numeric value?
15
votes
2answers
200 views

why $ \rm{lcm}[1,2,3,\cdots,n]\in (2^n,4^n)$

Let $n\ge 7$ be positive integers,show that $$f(n)=\rm{lcm}[1,2,3,\cdots,n]\in (2^n,4^n)$$ Anyone know this problem background?or maybe have best proof or best result?
1
vote
1answer
61 views

Perturbation theory - Algebraic equations (Repeated roots)

Obtain a three term approximation (for $E\to 0$) of the roots of the following equation: $$x^2 + (4+E)x + (4-E) = 0$$ I understand what to do in basic cases but I've no idea what to do in this ...
0
votes
1answer
18 views

Selecting $x_0$ in approximations using Taylor polynomials?

Are there any general rules for how to pick $x_0$ (relative to $x$) in approximations using Taylor polynomials? What if $x=x_0$?
1
vote
2answers
68 views

Is there a formula to approximate $\pi$ in the form of $\dfrac{p}{q}$?

Is there a formula which helps in approximation of $\pi$ as $\dfrac{p}{q}$ where $p,q \in \mathbb{Z}$? I got this site though : [http://qin.laya.com/tech_projects_approxpi.html ] which shows the ...
2
votes
0answers
32 views

How are Floating Point approximations done by integer operations? (Source Wikipedia)

Please help me understand the mathematics involved in Wikipedia page of Floating point, section of Piecewise Linear approximation to exponential and logarithm. Following is the link Piecewise linear ...
1
vote
3answers
67 views

How did Feynman do this approximation $\sqrt{A^2 + \mu^2\varepsilon^2} =A\biggl( 1+\frac{1}{2}\,\frac{\mu^2\varepsilon ^2}{A^2}\biggr)$?

I was reading the 9th lecture of Feynman; here is the excerpt of my concern: Let’s go back now to our particular example of the ammonia molecule in an electric field. Using the values for H11, ...
3
votes
0answers
93 views

Two-dimensional integral with variable inner limits

I'm a physics student trying to solve a scattering problem on the surface of a topological insulator and am currently stuck with integrals of this kind: $$\int_0^\pi d\phi \int_\phi^\pi ...
4
votes
3answers
121 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 ...
15
votes
1answer
316 views

Nested solutions of a quadratic equation.

A quadratic equation of the form $x^2+bx+c=0$ can be solved with the classical formula that gives all solutions. Here I want discuss some other methods to find one solution. The best known is by ...
0
votes
1answer
39 views

$x$ln$x$ quadratic approximation at $x=1$

$x$ln$x$ quadratic approximation at $x=1$ 2 ways : 1) quadratic approx whole thing : $(x-1)^2 + \dfrac {(x-1)^2} 2$ This isn't same as 2) split $x$ and take quad approx of ln$x$ at $x=1$ ...
1
vote
1answer
39 views

Related rates - approximation or not?

The radius of a uniform spherical balloon is increasing at 3% per second. Find the % rate at which its volume is increasing. My solution: Percentage is change of some value divided by that ...
0
votes
2answers
40 views

Is dv only approximate of dv/dx*dx?

I've tried to solve this problem: If $V=2x^{3}$ what is the approximate percentage change in $V$ when $x$ changes by 2%? My solution is: Since $x$ changes by 2% then $dx=0.02x$. ...
1
vote
2answers
46 views

Linearize differential equation

Given the following differential equation: $$ \dot{y}_1 + p(t)f(y_1) = g(t) $$ then I want to linearize it in the sense that it becomes a system of equations on the form: $$ \dot{y}_1 + p_1(t)y_1(t) = ...
0
votes
0answers
13 views

Solving or approximating the functions for implicit differential equations given by Euler's angles

Given that $\omega_x$, $\omega_y$ and $\omega_z$ are all easily integrable polynomial functions of time, and satisfy the equations: $\omega_x=\dot{\phi}\cdot \sin(\theta)\cdot ...
3
votes
2answers
70 views

Estimating numerically $\lim \limits _{x \to 0} \frac {\sin 4x} x$

As the title says, I am looking for ways to estimate numerically $\lim \limits _{x \to 0} \frac {\sin 4x} x$. So far, I've tried filling in numbers on either end of zero to make an estimate, and keep ...
7
votes
0answers
128 views

Approximating intervals and squares by increasingly dense disjoint finite sets with special properties

Apologies for the length of the question. Consider interval $I=[0,1]$. For any $n \in \mathbb{N}$ we can always find two finite sets $S_{1n} \subset I$ and $S_{2n} \subset I$ such that: a) ...
0
votes
1answer
61 views

Estimate an error using method similar to Stirling's approximation?

In the application of WLLN, which is the polynomial approximation. For any function $F\in C([0,1])$ can be approximated by a polynomail $G$ so as to make $||F-G||=\max_{0\le x \le 1}F(x)-G(x)$ as ...
1
vote
2answers
77 views

Stirling's Approximation for binomial coefficient

In this proof, it is assumed that, for $k << n$, ${n \choose k} \approx \frac{n^k}{k!}$, given Stirling's approximation. How does Stirling's Approximation, in either form $\ln n! \approx ...
1
vote
0answers
35 views

Solving of numerical equation with integrals

Let's have equation $$ \cosh(2 \pi x) = \cos\left[\text{Re}\int \limits_{0}^{2 \pi}\sqrt{A - 2q\cos(2z)}dz \right]\times \cosh\left[ \text{Im}\int \limits_{0}^{2 \pi}\sqrt{A - 2q\cos(2z)}dz\right], $$ ...
0
votes
1answer
25 views

Approximating basic trigonometric functions to a integrable form

I wondered if there is a way of approximating trigonometric functions in terms of basic functions (possibly trigonometric functions) so that one can derive the indefinite integral of said function. ...
13
votes
9answers
376 views

How do I Approximate $\log{2}\approx 0.693$ without using the Maclaurin series?

How do I approximate the value $\log{2}\approx 0.693$ without using the Maclaurin series? The book gives the hint: consider $f(x)=e^x-e^{-x}-2x$.
6
votes
0answers
267 views

Response Surface Methodology using Moving Least Squares Method

I would like to obtain the response surface of a mathematical function for reliability-based design optimization (RBDO). To obtain a reliably response surface, I learned that moving least squares ...
0
votes
0answers
14 views

Sobolev space for classic function approximation?

Hi guys could be any convenience in using a sobolev space instead of square integrable space for function approximation? I know that sobolev space are mostly used for PDE, but i was wandering if ...
3
votes
0answers
34 views

Solution for 4th grade polinomial equation

I'm development a physics model that require a expression for elongation of a elastic material, $\lambda=\frac{L}{L_o}$ [where $L$ is the thickness of the material and $L_o \equiv L(\sigma = 0)$] as ...
1
vote
1answer
28 views

ODE with time-dependent frequency

Let's have equation $$ \tag 1 \frac{d^{2}y(t)}{dt^{2}} +\omega^{2}(t)y(t) = 0, \quad t \in (t_{\text{in}}, \infty) $$ Here $$ \omega^{2}(t) = A(t) - B(t)cos(2t), $$ and functions $A(t), B(t)$ have ...
2
votes
2answers
39 views

Approximation of $(x+cx^2)/(1+cx^2)$, when c is small

I'm reading a paper and can't wrap my head around the following approximation: $f(x) = \frac{x+cx^2}{1+cx^2}$ $,$ $0 \le x \le 1$ Assuming that $c$ is small, $(c << 1)$, the following ...
0
votes
1answer
10 views

Function for approximating the definite integral of a function using an r-degree polynomial

We have the Midpoint Rule which approximates the definite integral of a function $f(x)$ over $[a, b]$ using $n$ sub-intervals with width $\Delta x$ using a degree-0 polynomial $A$: ...