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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2answers
74 views

Approximation of a quotient that involves the Lambert function.

I would like to find an asymptotic upper bound for $$\frac{-\ln n}{W(- \ln^{-c}n)}$$ where $c$ is positive and $W$ is the Lambert function. More precisely, I want something which dominates this ...
0
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0answers
15 views

Upper bound on the remainder of a polynomial (not taylor)

There are many ways of approximating a function with a polynomial, $\widehat{f}(x)\approx f(x)$. One way is the taylor polynomial. A nice property that goes along with the taylor polynomial is an ...
0
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1answer
102 views

Approximate roots of nonlinear equation (non-integer polynomial)

In case of pulsating bubble arising from underwater explosion, bubble radius satisfies the following equation. $x^3\dot{x}^{2} + x^3 + \frac{k}{x^{3(\gamma-1)}} = 1$ The minimum and maximum bubble ...
1
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2answers
23 views

What does it mean by the approximation $\int_a^bf(x)dx\approx\sum_{i=0}^nA_if(x_i)$ is exact for all polynomials of degree up to $2n+1$?

There is these notes about Gaussian Quadrature and I am trying to understand what does the sentence "is exact for all polynomials of degree up to $2n+1$" actually mean. Gaussian Quadrature - General $...
1
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2answers
21 views

A couple of inequality / similarity that don't make sense to me.

I was reading thru the proof for a combinatorics problem, but there were a couple places in there that gave me pause. In particular, one part of the proof had the following: $${n \choose t}\frac{2}{2^{...
1
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0answers
23 views

Alpha Max Plus Beta Min Calculation

I read about the Alpha Max Plus Beta Min algorithm described here. Here is a screenshot from the wikipedia page: I think understand what the algorithm is supposed to do. It makes an approximation ...
0
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0answers
28 views

How one can approximate irrational raised to irrational power?

How one can evaluate irrational number raised to irrational power? Like is there an easy way to prove that $-0.685<\pi^e-e^\pi<-0.675$?
1
vote
2answers
46 views

How do I find arc length using the trapezoid rule?

The question asks, "Use the trapezoid rule (when $n=8$) to approximate the arc length of the graph of $y=2x^3-2x+1$ from $A (0,1)$ to $B(2,13)$" I first graphed this out and found the points to have $...
0
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4answers
62 views

What is the name of the approximation $ \left(1-\frac{1}{x}\right)^n \approx e^{-n/x} $?

Which approximation allows for the following? $$ \left(1-\frac{1}{x}\right)^n \approx e^{-n/x} $$ Here both $x$ and $n$ are variables.
0
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0answers
16 views

Summation of shifted sigmoids

I recently came across this statement in a paper, and I was not able to figure out how to prove it. Any ideas would be appreciated. $$\sum_{i>0} \phi(x-i+0.5) \approx \log(1+e^x) $$ where $$\...
0
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0answers
7 views

Approximation for minimum and multiple product

Is there an approximation for the following formula: $min\{1,w+1-\left((1-x )\left(\prod\limits_i(1-y_i)(1-z_i)\right)\right)\}$ where $0 \lt w,x, y_i, z_i \le 1$.
0
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2answers
54 views

Approximate $\log(1-e^x)$ where $x<0$

The title is pretty self-explanatory, I need to calculate the logit function ($x=\log(p)$): $$x-\log(1-e^x)$$ Where $x<0$, And my problem is to approximate $$\log(1-e^x)$$ I was thinking of ...
0
votes
2answers
37 views

Error of Stirling’s approximation for Binomial with central limit theorem

So the question asks: Let $X_n$~Bin(2n,1/2),use Stirling’s approximation for $n!$ to show $P [X_n = n]$~ $1/√(πn)$ as $n→ ∞$, and show the error in the estimate for $P [X_n ≤ n]$, given by the central ...
0
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0answers
9 views

Expand or approximate entropy of a two-term Gaussian Mixture

Is it possible to create some expansion to approximate this $h(a)$ for $a>0$ near $a\rightarrow0$? $$N(x,v)\equiv\frac{1}{\sqrt{2\pi v}}e^{-\frac{x^{2}}{2v}}$$ $$ f(x,a)\equiv(1-a)*N(x,1)+a*N(x,10)...
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0answers
21 views

Approximation of function in a given interval

I need help with the following approximation problem. Given the orthogonal system of functions in $[-\pi,\pi]$. How do I approximate the function $f(x)=-1$, if$ -\pi\leq x\leq 0 $ and $f(x)=1$, if $ ...
0
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1answer
33 views

find monotonic, increasing function going exactly throught set of points

This questions stems from a problem that I encountered while writing a program. This problem was identical to this one. What I did on the spot was described as a "trick" with binary search over ...
0
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0answers
21 views

Effect of location of nodes for interpolation

I've been doing some numerical experiments to see how the location of the interpolating nodes affects the performance of the interpolator. I am just curious about this because it seems like the ...
0
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0answers
13 views

Partial Fourier Series Error

I calculated the coefficient of the complex fourier series of a trapezoidal signal to be: $$c_n = \frac{\tau}{T} \frac{sin(0.5n\omega_0\tau) \cdot sin(0.5n\omega_0\tau_r)}{(0.5n\omega_0\tau) \cdot (0....
0
votes
0answers
46 views

How to improve Euler-Maruyama discretisation with analytical moments?

I'm trying to improve Euler-Maruyama discretisation by adding to it the analytical moments. To try it I made a very simple example on the stochastic process $X(t) = W(t)^2$, where $W(t)$ is a standard ...
0
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1answer
23 views

Choosing the knots for a linear interpolation

I want to approximate a function through piecewise linear interpolation and try to understand how I could set the associated interval points optimally. Take a continuous function $f: X \rightarrow \...
3
votes
0answers
22 views

What's the most accurate way to estimate a percentile from multiple partial percentiles?

There exists 3 sets of numbers. I have the 99th percentile (p99) of each set and the cardinality of the set, but not the values in the set themselves. p99: 540, cardinality: 215 p99: 288, ...
2
votes
2answers
71 views

Upper bound on ratio of incomplete Gamma function and Gamma function $\frac{ \Gamma \left( x; a\right)}{\Gamma(x)}$

I am trying to find a tight upper bound the following expression \begin{align} \frac{ \Gamma \left( x; a\right)}{\Gamma(x)} \end{align} where $\Gamma \left( x; a\right)$ is incomplet Gamma function \...
0
votes
1answer
40 views

Least positive integer such that $\cos^k \left(\frac{\pi}{2k}\right)\geq \frac{99}{100}$

I wrote a program to figure this out and found $k=123$. Writing $f(k)$ as the function, I showed that $f$ is strictly increasing on the positive integers, and Wolfram agrees that $f(122)<99/100<...
1
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2answers
73 views

Is there integral or series for $\sqrt{10}-\frac{4^4}{3^4}$ (to prove the inequality)?

Both of these numbers are bad approximations for $\pi$, but they turn out to be much closer together: $$\sqrt{10}-\frac{4^4}{3^4}=0.00178$$ Since there is a lot of questions here about integrals and ...
0
votes
1answer
44 views

How to find an approximation of power series to correct to within $10^{-7}$ as faster?

If I approximate a $\displaystyle\int_{0}^{0.5} \frac{1}{1+x^7} dx$ correct to within $10^{-7}$. How to find it without using a calculator? Now I can't. I usually calculate every single term, and ...
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0answers
12 views

The Wronskian of parabolic cylinder function and the plane wave

Suppose equation $$ \tag 1 \ddot{y} + (t^2\theta (t - t_{i}) + p^2)y(t) = 0, \quad t \in (t_{0}, \infty) $$ (here $\theta (t - t_{i})$ is the step function) with initial condition $$ \tag 2 y(t \to t_{...
0
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0answers
5 views

Determine $(g \circ f)'(1, 1)$ and an approximate value of $(g \circ f)(1,01; 1,01)$. Approximation by differentials and chain rule.

First time posting. Excuse me for the formatting or grammar. Question Let $f: R^2 \to R^3$ be a differential function, such that $f(1,1) = (3, 1, 2)$ and $f'(1,1): R^2 \to R^3$ is given by the ...
4
votes
1answer
69 views

The 9 most significant digits in Fibonacci series (Project Euler 104)

With regards to project Euler, problem 104: https://projecteuler.net/problem=104 The essence of the question here is how to keep track of the 9 most significant digits of a Fibonacci series (Keeping ...
0
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1answer
31 views

approximation of binomial coefficient by exponentiation

Show that inequality holds for every n, k. $$\ { n \choose k} \leq \frac{n^k}{k!}. ( \, 1-\frac {k}{2n} ) \,^{k-1} $$ I used Stirling's formula but I stucked, which is $$\ { n \choose k} = \frac{n^...
-1
votes
2answers
32 views

How to arrive at this approximation? [closed]

I encountered an equation: $$\frac{1}{(ja + \delta{z_{n}} - \delta{z_{n-j}})^2} + \frac{1}{(-ja + \delta{z_{n}} - \delta{z_{n+j}})^2}$$ can someone tell me how it approximates to: $$-2\left[\frac{\...
1
vote
1answer
31 views

Stuck in using Stirling's approximation to show and justify an approximation of the number of permutations with and without ordering

This is a problem from my applied mathematics class where we are currently working on using Stirling's approximation which is: $ n! \sim (\frac{n}{e})^n \sqrt{2 \pi n} $ and the context of this ...
2
votes
3answers
57 views

Rational approximation of square roots

I'm trying to find the best way to solve for rational approximations of the square root of a number, given some pretty serious constraints on the operations I can use to calculate it. My criteria for ...
2
votes
0answers
55 views

Does the continued fraction for $e^{3/n}$ have a pattern?

Wikipedia has patterns for the simple continued fractions $e^{1/n},e^{2/n}$, which made me wonder whether there is one known for $e^{3/n}?$ (by pattern, I mean that the partial quotients $a_n$ can ...
1
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1answer
22 views

propagation of error from product of Taylor Series

Say I have two functions $f(x)$ and $g(x)$, both of which I will be approximating with Taylor series $T_f(x)$ and $T_g(x)$ respectively. Lets say $f(x)$ is order $O(x^{n_1})$ and $T_f(x)$ has error of ...
1
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0answers
19 views

Approximating number of partitions of $n$, denoted $p(n)$, by $p(n)\ge e^{c\sqrt n}$

I was to show that $p(n)$, the number of partition of a positive integer $n$, satisfy: $p(n)\ge \max_{1\le k\le n}{{n-1\choose k-1}\over k!}$, which was obvious because every unordered collection of ...
1
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1answer
38 views

Linear Approximation for functions with first derivative as $0$

Linear approximation around a point through Taylor series requires the first order derivative to be non-zero unless you want to get only the value at that point. However this is only true when you are ...
2
votes
2answers
42 views

Approximating $f(1)$ given $f(0)=2$ and $\frac{1}{2}x^2≤f'(x)≤x^2$?

This is an exercise using the mean value theorem: Approximate $f(1)$ given $f(0)=2$ and $\frac{1}{2}x^2≤f'(x)≤x^2$ for $x≥0$. I've found (using MVT): $$\frac{1}{2}x^2+2≤f(1)≤x^2+2$$ and I can ...
3
votes
2answers
25 views

Minimal Subset that sums up to

Let $X \subsetneq \mathbb{N}$ be a finite set, and $c \in \mathbb{N}$ we are looking for a subset $$ Y \subseteq X $$ such that $\sum_{y \in Y} y \geq c$. Assuming a subset that satisfies the ...
2
votes
3answers
110 views

Approximating $\exp(\sin(x))$ with polynomials

I want to find explicit formula for the sequence $f_n$ of polynomials which uniformly convergent to $\exp(\sin(x))$ on $[0,2014]$. Taylor's expansion is terrible for this function so i think that ...
12
votes
1answer
227 views

Where am I violating the rules?

Being fascinated by the approximation $$\sin(x) \simeq \frac{16 (\pi -x) x}{5 \pi ^2-4 (\pi -x) x}\qquad (0\leq x\leq\pi)$$ proposed, more than 1400 years ago by Mahabhaskariya of Bhaskara I (a ...
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0answers
31 views

Help with basic arithmetic involving Big Oh

I'm trying to determine the resulting "Big Oh" when arithmetic operators are applied between two different functions, but I'm a bit unsure after looking at even the basic operators shown on wikipedia ...
0
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0answers
27 views

Upper-bounding $\mathbb{E}\left[ \frac{\tilde{p}}{p} a \right]$ with $D(p, \tilde{p})+ \mathbb{E}[a]$?

Define two slightly different probability distributions: $$ p_k = \frac{ \exp \left[ - \eta L_k \right] }{ \sum\limits_{i=1}^K \exp \left[ - \eta L_{i} \right] }, \quad \tilde{p}_k = \frac{ \exp \...
0
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0answers
18 views

What “approximation factor” means for approximation algorithms?

The paper I read starts with the following text: One of the most appealing open problems in the graph streaming area (see Problem 60 in [1]) is to close the gap between the approximation factor (i....
3
votes
1answer
56 views

Deriving Stirling's approximation formula via the definition of the Gamma function

In my asymptotic analysis and combinatorics class I was asked this question: We first remember the definition f the Gamma function $ \Gamma(n+1) = n! = \int_{0}^{\infty} t^{n} e^{-t} dt $ and ...
8
votes
2answers
850 views

Sum of inverse of Fibonacci numbers

If $F(n)$ is the nth Fibonacci number, How can I prove that: $$\sum_{i=1}^{\infty} \frac{1}{F(i)}\approx 3.36\, .$$
16
votes
2answers
221 views

Elegantly Proving that $~\sqrt[5]{12}~-~\sqrt[12]5~>~\frac12$

$\qquad$ How could we prove, without the aid of a calculator, that $~\sqrt[5]{12}~-~\sqrt[12]5~>~\dfrac12$ ? I have stumbled upon this mildly interesting numerical coincidence by accident, ...
0
votes
1answer
51 views

Limit of regular polygons approaching pi - earliest proofs

Archimedes used areas of regular polygons to approximate pi. He calculated both inner and outer polygons and realized that more sides yielded closer results to each other. There's surviving proof that ...
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0answers
26 views

Constraint on curvature of polynomial approximation for camera calibration

OpenCv uses the following polynomial for estimating lens distortion. More information can be found here. $\frac{1+k_1r^2+k_2r^4+k_3r^6}{1+k_4r^2+k_5r^4+k_6r^6}$ There is a lot more to the ...
2
votes
3answers
50 views

How to show that $2 ^\binom{N}{2} \sim \exp(\frac{N^2}{2}\ln(2))$

How can we show that: $2 ^\binom{N}{2} \sim \exp(\frac{N^2}{2}\ln(2))$ for large N.
1
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0answers
34 views

$\int_{-\pi/2}^{\pi/2} |\vec{v}-\vec{V}|(\vec{v}-\vec{V}) d t$, with $\vec{v}=(\vec{i} \cos t +\vec{j} \sin t) (\sqrt{1-p^2 \cos^2 t}+p \sin t)$

I need to solve this integral: $$\int_{-\pi/2}^{\pi/2} |\vec{v}-\vec{V}|(\vec{v}-\vec{V}) d t$$ $$\vec{v}=(\vec{i} \cos t +\vec{j} \sin t) \sqrt{1-p^2 \cos 2 t+2 p \sin t \sqrt{1-p^2 \cos^2 t}}=$$ $...