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
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
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 ...
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
48 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
46 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 ...
0
votes
1answer
11 views

Derive a second order difference approximation

Derive a second order difference approximation to $y(a)$ using the values $$y(a + h/2), \space y(a + h) \space,\space y(a + 2h)$$ Verify the order of your approximation. Have no idea how to tackle ...
0
votes
0answers
28 views

Approximately identical distributions

Suppose we have two random variables $x,y$ in $\mathbb{R}^n$. Assume that for some scalar $\epsilon>0$, for any set $S\subset\mathbb{R}^n$ there exists sets $S_1\supset S,S_2\supset S$ such that ...
14
votes
1answer
457 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
22 views

Need help in understanding appoximation for heavy hitters problem

I am reading a paper and do not understand the following "We allow the space used by a solution to grow as $1/ \epsilon $, so as $ \epsilon ↓ 0$ the space blows up..." I do not understand the ...
2
votes
1answer
47 views

Error Bounded Cubic B-Splines with fewest segments

I have some odd constraints in my project. Suppose we want to use Cubic BSplines to approximate a set of Points. There is two Constraints: error value should be an input to the algorithm (error ...
1
vote
1answer
17 views

Asymptotic approximation to radial wave function

This seemingly easy analysis is driving me up the wall. $$\frac{\text{d}^{2}u}{\text{d}\rho^{2}}=\left[1-\frac{\rho_{0}}{\rho}+\frac{l(l+1)}{\rho^{2}}\right]u$$ why is it for $$\rho\rightarrow0,$$ ...
0
votes
0answers
18 views

Softmax for Continuous Functions?

The softmax $\log \sum_{i=1}^n \exp(f_i)$ of vector $f$ is a smooth upper bound on $\max_i f_i$. However, the same cannot be said of $\log \int_{X} \exp(f(x))dx$ in relation to $\max_{x \in X} f(x)$ ...
1
vote
2answers
31 views

Approximation of $\frac{x}{\sqrt{x^2+R^2}}$

How do you prove this statement? If $x\gg R$ then $$\frac{x}{\sqrt{x^2+R^2}}\cong 1-\frac{1}{2}\left(\frac{R}{x}\right)^2$$ I have no ideas even how to start.
0
votes
1answer
38 views

Method for finding square roots quickly (manual)

I was recently studying AC circuits and there I need to use Pythagoras theorem a lot.So I was looking for a method with which square roots can be calculated very fast,manually up to 1 decimal ...
2
votes
1answer
62 views

Find a>1 s.t. $a^x = x$ has a unique solution

What $a$ makes $\{x\mid a^x = x\}$ a singleton? $$(1.4444)^x - x \le 0 \tag 1$$ has real solutions. $$(1.4447)^x - x \le 0 \tag 2$$ has no real solutions. I guess $1.4444 < a < 1.4447$ I ...
1
vote
0answers
25 views

Asymptotic behaviour of generalized binomial coefficient $\frac{an(an-1)…(an-n+1)}{n!}$

Let $a\in(0,1)$. What is the asymptotic behaviour of $\frac{an(an-1)...(an-n+1)}{n!}$ as $n\rightarrow\infty$? It looks like it might be possible to express this in terms of gamma functions and use ...
1
vote
1answer
33 views

Order of the error for the Trapezoidal and Simpson's method of numerical integration

What are the order of the error for the Trapezoidal and Simpson's method of numerical integration? What is the definition of order of the error of a quadrature formula? Is it true that order of ...
3
votes
1answer
28 views

Approximation formula on a surface

[Beginning calculus question.] We can get an approximation to the value of a function of two variables, I think, by saying $$ f(a+\Delta x , b+ \Delta y) \approx f(a,b) + f_x(a,b)\Delta x ...
2
votes
1answer
37 views

Approximating logarithmic series

I'm trying to solve the below recurrence $$T(n) = 2 T(n/2) + n/\log n$$ I referenced some online articles they all are approximating $n \sum_{i=0}^{h-1} \frac{1}{\log n - i}$ to $n ...
1
vote
2answers
70 views

Approximating a Harmonic Sum

The infinite sum $\sum_{n=1}^{\infty}\frac{1}{n}$ diverges. However, it is possible to find bounds from some $n$ to another integer $n$. Wolfram alpha is able to give a decimal approximation of the ...
2
votes
2answers
170 views

Approximation of $\pi$ using Brahmagupta's Identity

Brahmagupta, an ancient Indian Mathematician, gave an pretty efficient algorithm for finding integer solutions to the famous Pell's Equation, far before Fermat propounded this before the European ...
2
votes
1answer
34 views

Finding $w_1,w_2,b$ such that $w_2 \sigma(w_1 x + b) \approx x$

Let $\sigma(z) = 1/(1+e^{-z})$. How can I find $w_1, w_2, b$ such that $w_2 \sigma(w_1 x + b) \approx x$ for $x \in [0,1]$? The hint provided was to rewrite $x = \frac{1}{2}+\Delta$, assume $w_1$ is ...
1
vote
1answer
51 views

Probability that sum of two random variables is $1$

Let $X$ and $Y$ be two independent random variables with density functions $f_x(x) = x\exp\left(-\frac{1}{2}x^2\right) \mathbb{}$ where $x \in \mathbb{R}^+$ and $f_y(y) = \frac{1}{2}$ with $y \in ...
2
votes
0answers
29 views

Meaure-theoretic induction: Why dyadic approximation?

In measure-theoretic induction proofs we always use the dyadic approximation of a non-negative measurable function $Y$ as $$Y_n = \sum_{k=0}^{n2^n-1} k/2^n 1\left(\frac{k}{2^n} \leq Y < ...
1
vote
1answer
40 views

How are polynomials graphs approximated?

Say I have the data: $x=[ 1, 2, 3.3, 4, 5.5, 8, 9, 10.2, 11, 45 ]$ $y=[ 9,27,64,91,164,330,462,540,630,10218]$ The data is subjective though. How would one approximate a valid polynomial for this ...
5
votes
4answers
637 views

How can I calculate or at least approximate the sum?

As a part of a complexity analysis of the algorithm, I have to calculate the following sum: $$n^{1/2} + n^{3/4} + n^{7/8} + ...$$ where in total I have $k$ elements to sum: ...
0
votes
1answer
20 views

error bound for polynomial interpolation with derivative matching

We all know the following formula for the maximum error (evenly spaced) polynomial interpolation: $|f(x) - p_n(x)| \leq \frac{h^{n+1}}{4(n+1)} \max_{x\in [a,b]} f^{(n+1)}(x)$ where $p_n(x)$ is the ...
2
votes
1answer
47 views

Ways to generate triangle wave function.

I recently when searching for parameters on a unit cube in $\mathbb{R}^9$ (we all have our more or less peculiar hobbies, don't we?) found a practical reason to implement a triangle wave function ...
1
vote
0answers
28 views

Normalization in least-p'th minimax algorithm

In the book "Practical Optimization: Algorithms and Engineering Applications", the least-$p$th minimax algorithm is presented, for approximation of the minimax optimizer (Alg. 8.1): $Loss_x(k)$ = ...
2
votes
3answers
67 views

Bounds on Gaussian infinite sum

What are some good upper and lower bounds on the following sum? $$S=\sum_{n=-\infty}^{+\infty}\dfrac{1}{\sigma \sqrt{2 \pi}}e^{-\frac{1}{2}\left(\frac{n}{\sigma}\right)^2}$$ I am looking for ...
7
votes
3answers
465 views

After 6n roll of dice, what is the probability each face was rolled exactly n times?

This is closely related to the question "If you toss an even number of coins, what is the probability of 50% head and 50% tail?", but for dice with 6 possible results instead of coins (with 2 possible ...
2
votes
1answer
62 views

Exponential series approximation

I have a series of the following form: \begin{align} \sum_{k=2}^\infty \left( 1 - e^{-ns^{k-1}} \right)^k \end{align} where $0<s<1$. I would like to compute an approximation of this series, for ...
2
votes
3answers
56 views

Approximation by using Taylor Polynomials - why?

Could anyone tell me why would I want to approximate a function $f$ by using its Taylor expansion (is it the same as saying approximation by Taylor polynomials?), if I have the exact formula of the ...
1
vote
1answer
42 views

(Ab)using the factorial and gamma functions

I have a product of the following form: $$ P = (N-\alpha)(N-2\alpha)\cdots(N-k\alpha) $$ where $k$ is an integer between $1$ and $N-1$, and $\alpha$ is a real number in $(0,1]$. Clearly, for ...
0
votes
1answer
23 views

Minimizing sums of values versus minimizing cubes of sums.

I am attempting to find the best path from start to finish from a set of points. Say that one path has costs $x_1,x_2,...,x_n$ and $y_1,y_2,...,y_n$ associated with it. I am attempting to find the ...
1
vote
2answers
37 views

Complex Equation Formula

Can someone show me how the following two expressions are equivalent: $$\Gamma = \frac{i X - R_c}{i X + R_c} = -e^{-i 2 \mathrm{tan}^{-1} (\frac{X}{R_c})}$$ I'm working through a calculation and I ...
2
votes
0answers
27 views

Approximation of integrable function by polynomials

Assume $f\in \mathscr{R}(\alpha)$ on $[a,b]$, and prove that there are polynomials $P_n$ such that $$\lim \limits_{n\to \infty}\int_{a}^{b}|f-P_n|^2d\alpha=0.$$ Proof: Let $\varepsilon>0$ be given ...
-1
votes
2answers
33 views

Approximate a summation

Approximate $3+ \displaystyle \sum_{x = 2}^{999}\dfrac{3(1000-x)}{1000+x}$. It may help to know that $\ln 2 = 0.69$. I was thinking of doing the integral test to approximate this but I am unsure if ...
2
votes
3answers
57 views

Why does $p$ have to be moderate in the Poisson approximation to binomial random variable?

So the proof that a binomial rv with large $n$ approximates a poisson rv with $\lambda = np$ (given below) doesn't seem to use the fact that $p$ is moderate/small, so why does wikipedia and my ...
3
votes
3answers
63 views

Numerical method for approximating the standard Normal distribution cdf with mean 0 and variance 1

The standard Normal distribution probability density function is $$p(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2},\int_{-\infty}^{\infty}p(t)\,dt = 1$$ i.e., mean 0 and variance 1. The cumulative distribution ...
0
votes
1answer
34 views

Interpolation and divergence of $n$-th derivative

i have a curiosity. Let's say i have a function $f \in C^{\infty}[a,b]$ such that there's a $x_0 in [a,b]$ that makes and $f(x_0) = 0$ and $a_n = f^{(n)}(x_0)$ diverges (i.e. $|a_n|=\infty$) could ...
-1
votes
1answer
31 views

Solution to $0= c_1 x+c_2 x \ln \left(\frac{1+x}{x} \right)+c_3 \ln\left(\frac{1+x}{x} \right)+c_4$

We want to solve \begin{align} 0= c_1 x+c_2 x \ln \left(\frac{1+x}{x} \right)+c_3 \ln\left(\frac{1+x}{x} \right)+c_4 \end{align} for $x \in [0,1]$ and where $c_1,c_2,c_3,c_4$ are non-zero constants. ...
10
votes
2answers
275 views

Proof of $\sum_{n=1}^{\infty} \frac{x^n \log(n!)}{n!} \sim x \log(x) e^x$ as $x \to \infty$

Prove that $$\sum_{n=1}^{\infty} \frac{x^n \log(n!)}{n!} \sim x \log(x) e^x \,\,\,\text{as}\,\,\, x \to \infty$$ and $$\sum_{n=1}^{\infty} \frac{(-x)^n \log(n!)}{n!} \to 0 \,\,\,\text{as}\,\,\, x \to ...
2
votes
1answer
158 views

Approximating on a line

Say I have sampled some points in $[0,1]^2$ and evaluate a function $f(x,y)$ for them. I am interested in the behavior of $f$ along a single dimension. If the points were like ...
2
votes
1answer
44 views

How do you Taylor expand the log likelihood function of the Poisson distribution?

This question is an extension to this previous question asked by myself: When doing a maximum likelihood fit, we often take a ‘Gaussian approximation’. This problem works through the case of a ...
2
votes
0answers
22 views

For $\sum_{n=1}^\infty (-1)^n\frac{(x-3)^n}n$ approximate when x = 4, with an error that does not exceed .01

For $\sum_{n=1}^\infty (-1)^n\frac{(x-3)^n}n$ approximate with specific details the series when x = 4, but with an error that does not exceed .01. That is, find a value of n so that the nth partial ...
1
vote
0answers
34 views

Approximating $|1-e^{i\delta}|$

Actually this question is derived from the Rudin's Real and Complex Analysis's lemma 15.17. In the lemma, given condition is below; Let $h(z) \in H(\Omega)$ such that Re $h(z) = \log |1-z|$, |Im ...