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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7
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3answers
609 views

How is it that this shape can converge to what looks like a triangle but has a different perimeter?

I had this strange notion some time ago, and I recently wrote a blog post about it, as a mere curiosity. I don't really consider it a "serious" mathematical question; but out of interest, I wondered ...
6
votes
1answer
233 views

Computer Algebra Systems for Experimental Mathematics (especially Integer Relations with PSLQ)

I would like to use a computer algebra system to do some experimental mathematics, particularly Integer Relation problems using the PSLQ algorithm. I know that Maple has a PSLQ implementation, but ...
5
votes
3answers
899 views

How to find an approximation to $1 - \left( \frac{13999}{14000}\right )^{14000}$?

I want to find an approximation to the expression $$ 1 - \left( \frac{13999}{14000}\right )^{14000} $$ I tried by taking logarithm $$ \ln P = \ln\left(1 - ...
4
votes
3answers
110 views

Symmetry Of Differentiation Matrix

I have a problem computing numerically the eigenvalues of Laplace-Beltrami operator. I use meshfree Radial Basis Functions (RBF) approach to construct differentiation matrix $D$. Testing my code on ...
4
votes
4answers
205 views

How to compute a lot of digits of $\sqrt{2}$ manually and quickly?

After having read the answers to calculating $\pi$ manually, I realised that the two fast methods (Ramanujan and Gauss–Legendre) used $\sqrt{2}$. So, I wondered how to calculate $\sqrt{2}$ manually in ...
4
votes
0answers
259 views

Differences among Cauchy, Lagrange, and Schlömilch remainder in Taylor's formula: why is generalization useful?

I would like to know what really are the main differences (in terms of "usefulness") among Cauchy, Lagrange, and Schlömilch's forms of the remainder in Taylor's formula. Could you provide examples ...
4
votes
2answers
554 views

Method of dominant balance and perturbation

Approximate the solutions of $$\epsilon x^4 + (x-1)^3=0$$ I can't perform a singular perturbation because if I let $\epsilon=0$ then I lose a root. My professor suggests The Method of Dominant ...
4
votes
1answer
101 views

$f$ is approximated uniformly on $R$ by $p_n(x)$, then $f$ is a polynomial

Suppose $p_n(x)$ is a sequence of polynomials which converge to a function $f$, uniformly on $\mathbb{R}$. Show that $f$ is a polynomial. If there were a uniform bound $M$ on the degree of $p_n(x)$, ...
3
votes
1answer
325 views

What is the proof of the rules of significant figures?

I wanted to know how do we know that the rules that we follow when doing arithmetic with significant figures are correct? Like why when adding or subtracting we keep the same number of decimal places ...
3
votes
2answers
2k views

Approximations for the partial sums of exponential series

Though the question here (Partial sums of exponential series - Stack Exchange) is similar, it is more specialized and I rather need a general approximation for an arbitrary partial sum. Essentially, ...
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
1answer
180 views

Existence of a simultaneous rational approximation of real numbers in (0,1)

I have a simple question the rational approximation of real vectors. Dirichlet's simultaneous approximation theorem states: Given any $d$ real numbers $\alpha_1,\ldots,\alpha_d$ and for every ...
2
votes
1answer
133 views

Chebyshev coefficients from a polynomial

Is there an efficient algorithm for finding the coefficients in a Chebyshev basis of a polynomial? That is, given the set of $a_k$ such that: $p_n(x) = \sum_{k=0}^N a_k x^k $ Find the set of $c_k$ ...
1
vote
2answers
616 views

Product of Gamma functions II

What is the value of the product of Gamma functions \begin{align} \prod_{k=1}^{10} \Gamma\left(\frac{k}{10}\right) \end{align} and can it be shown that \begin{align} \prod_{k=1}^{20} ...
1
vote
2answers
1k views

Approximation in $L^2$ by piecewise constant functions

Dear all, I'd like to know if there is any general result on the approximation of $L^2$ functions by piecewise constant functions. More specifically, I'd like to know if the following approximability ...
0
votes
1answer
239 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 ...
7
votes
4answers
8k views

How to simplify or calculate a formula with very big factorials

I'm facing a practical problem where I've calculated a formula that, with the help of some programming, can bring me to my final answer. However, the numbers involved are so big that it takes ages to ...
6
votes
5answers
1k views

What exactly is “approximation”?

There are a lot of great "approximations" that exist in the mathematical field:$$\dfrac{22}{7} \approx \pi$$ $$e \approx \left(1 + \dfrac{1}{n}\right)^n$$But the fact that I have yet to know what ...
6
votes
3answers
45k views

Approximation symbol: Is π ≈ 3.14.. equal to π ≒ 3.14..?

This could be a trivial question, but what is exactly the difference of between these two expressions? Am I correct to state the both interchangeably whenever I need to express the approximation of ...
5
votes
0answers
66 views

Pointwise approximation of a closed operator

If $T:\mathcal D(T) \rightarrow \mathcal Y$ is a closed operator from a Banach space $\mathcal X$ to a Banach space $\mathcal Y$, is it possible to find bounded operators $T_n\in \mathscr B(\mathcal ...
4
votes
4answers
89 views

Generalization of piece-wise linear functions over a metric space

Suppose we want to construct a function $f$ from a compact metric space $(X,\rho)$ to a Euclidean space $\mathbb{R}^n$ that is Lipschitz continuous with a constant $L$: $$ \forall x,y \in X . ...
4
votes
3answers
1k views

How to calculate $\exp(-x)$ using Taylor series

We know that the Taylor series expansion of $e^x$ is \begin{equation} e^x = \sum\limits_{i=1}^{\infty}\frac{x^{i-1}}{(i-1)!}. \end{equation} If I have to use this formula to evaluate $e^{-20}$, how ...
4
votes
1answer
226 views

Evaluating a limit involving binomial coefficients.

If $N_c=\lfloor \frac{1}{2}n\log n+cn\rfloor$ for some integer $n$ and real constant $c$, then how would one go about showing the following identity where $k$ is a fixed integer: $$\lim_{n\rightarrow ...
3
votes
1answer
86 views

Inverse of prime counting function

The prime counting function $ \pi (x) \approx \dfrac {x} {\ln(x-1)} $. This function returns the number of primes less than $x$. Note: $x-1$ gives a better estimate than $x$. How to find $x$ given $ ...
3
votes
1answer
84 views

Does projection onto a finite dimensional subspace commute with intersection of a decreasing sequence of subspaces: $\cap_i P_W(V_i)=P_W(\cap_i V_i)$?

Let $V$ to be an infinite dimensional linear space over some field $k$. (you can take $k=\mathbb{C}$, or further assume $V$ is a complex Hilbert space). And assume $W$ is a finite dimensional ...
3
votes
2answers
785 views

How to bound the truncation error for a Taylor polynomial approximation of $\tan(x)$

I am playing with Taylor series! I want to go beyond the basic text book examples ($\sin(x)$, $\cos(x)$, $\exp(x)$, $\ln(x)$, etc.) and try something different to improve my understanding. So I ...
3
votes
2answers
443 views

Asymptotic number of unlabeled graphs

A rather tight lower bound $c(n)$ of the number of unlabeled digraphs of order $n$ (loops allowed) seems to be $$c(n) = 2^{n^2}/n!$$ because there are $2^{n^2}$ labeled graphs, almost all of them ...
3
votes
1answer
678 views

Approximation for Lambert W function near zero

I am looking for a good approximation for the $W_0$ branch of the Lambert $W$ function. I am looking for values $0 < x < e$ only, so I expect something simpler than the general Taylor expansion. ...
2
votes
5answers
81 views

Is there any good approximation for $\prod_{i=3}^k (n-i)$? $(n \gg k)$

Is there any good approximation for $\prod_{i=3}^k (n-i)$? $(n \gg k)$ I know that $\prod_{i=3}^k (n-i) < \prod_{i=3}^k n = n^{k-2}$ Also a tighter upper bound is appreciated.
2
votes
1answer
722 views

Approximating a large number of data points using (cubic) splines in l1/l2 norm.

I have a pretty large dataset ($x,y$) consisting of a few million points. There is a lot of noise in the data. I want to find a smooth but simple approximation/representation for this dataset, so that ...
2
votes
1answer
723 views

Bounding the modified Bessel function of the first kind

i'm looking for an upper bound for the modified Bessel function of the first kind of a +ive real argument. It seems that it satisfies the inequality : $$I_{n}(x)\leqslant \frac{x^{n}}{2^{n}n!}e^{x}$$ ...
2
votes
1answer
54 views

Prove that s is finite and find an n so large that $S_n$ approximate s to three decimal places.

$$S=\sum_{k=1}^\infty\left(\frac{k}{k+1}\right)^{k^2};\hspace{10pt}S_n=\sum_{k=1}^n\left(\frac{k}{k+1}\right)^{k^2}$$ Let $S_n$ represent its partial sums and let $S$ represent its value. Prove that ...
2
votes
1answer
511 views

Approximating Stirling's number of the second kind to allow for large inputs

I'm looking for an approximation for Stirling's number of the second kind, $S_2(n,k)$, which counts the ways to partition a set of $n$ objects into $k$ non-empty subsets: ( ...
1
vote
1answer
57 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 ...
1
vote
2answers
131 views

Can every continuous function on complex domain be approximated by polynomials pointwise?

Do you know any theorem that will help me with this question: Let $f$ be any continuous function on complex plane. Show that there is a sequence $(P_n)$ of polynomials such that $P_n$ converges ...
1
vote
1answer
2k views

How to approximate an integral using the Composite Trapezoid Rule

I'm trying to estimate the value of the following integral on the interval $[0,1]$ $$ I = \int_0^1 \frac{1}{1+x} dx $$ So, using the composite trapezoid rule (and with $n=4$, ie I'm only using the ...
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 ...
0
votes
1answer
45 views

a question on decreasing sequence of subspaces II

This is related to this question see here Let $V= \mathbb{Q}^{\mathbb{Z}}$, considered as an infinite dimensional linear space over $\mathbb{Q}$. And assume $W=\mathbb{Q}^F$ is a finite dimensional ...
0
votes
3answers
814 views

Approximating the integral $\int_0^{0.1} \sqrt{1-1/2\sin^2(t)} dt$

How would I go about evaluating $I = \int_0^{0.1} \sqrt{1-1/2\sin^2(t)} dt$ to four decimal accuracy?
0
votes
1answer
286 views

Approximate solution for an exponential equation

Trying to solve this question: Probability of ball ownership I got at an expression for the solution, P: $$\frac{P}{M} = (1 - \frac{1}{M+N})^{N(1-\frac{P}{M})+M}$$ Where M, N are parameters. The ...
5
votes
1answer
90 views

Inverse of $\frac{1-e^{-x}}{x}$ on $(0,1)$

I am trying to invert (or to estimate the inverse of) $$y=\frac{1-e^{-x}}{x}$$ for $y\in(0,1)$. The function 'looks' monotonically decreasing between $x=0$ and $x=\infty$, but I have not been able to ...
5
votes
5answers
3k views

Numerically Efficient Approximation of cos(s)

I have an application where I need to run $\cos(s)$ (and $\operatorname{sinc}(s) = \sin(s)/s$) a large number of times and is measured to be a bottleneck in my application. I don't need every last ...
4
votes
2answers
112 views

Approximation of DE

It depends on my previous question. Closed form solution of DE I don't want to deal with Airy functions. How can I approximate this DE in continous domain $[0,1]$? $$y''(x)+(x+1)y(x)=0\quad\text{ ...
4
votes
2answers
3k views

Modern formula for calculating Riemann Zeta Function [duplicate]

Possible Duplicate: How to evaluate Riemann Zeta function I have an amateur interest in the Zeta Function. I have read Edward's book on the topic, which is perhaps a little dated. I would ...
4
votes
1answer
529 views

Hermite Interpolation of $e^x$. Strange behaviour when increasing the number of derivatives at interpolating points.

I am trying to understand Hermite Interpolation. Here is my pedagogical example. I want to approximate $f(x)=e^x$ on the domain $[-1,1]$ using Hermite interpolation. I choose the Chebyshev zeros ...
3
votes
2answers
143 views

Bounds on $ \sum\limits_{n=0}^{\infty }{\frac{a..\left( a+n-1 \right)}{\left( a+b \right)…\left( a+b+n-1 \right)}\frac{{{z}^{n}}}{n!}}$

I have a confluent hypergeometric function as $ _{1}{{F}_{1}}\left( a,a+b,z \right)$ where $z<0$ and $a,b>0$ and integer. I am interested to find the bounds on the value it can take or an ...
3
votes
1answer
1k views

With Euler's method for differential equations, is it possible to take the limit as $h \to 0$ and get an exact approximation?

I was recently watching a tutorial on Euler's method for approximating differential equations, and the whole time I was thinking "why can't you just take the limit of the step size $h$ as it goes to ...
2
votes
0answers
81 views

Why the quadrature formula is exact one not an approximation?

I am reading this material on the algorithm of calculating the centroid of a polyhedron. I am confused by the last step of the deduction: The three coordinates of the centroid can be obtained: ...
2
votes
4answers
228 views

How to extract fraction from a floating point number

I'm making some tests with float type (floating point number) with programming and in some of my tests I need to extract the fraction that originates the float value. Let $ x $ be a floating point ...
2
votes
2answers
42 views

Least Squares approximation for item prices

Let's say that $A$, $B$, $C$ are different items with different values. $R$ is a unit of currency, for simplicity I'll let it be $1$. Traders frequently trade these items on an open market. Price is ...