Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [approximation]

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).

Approximations are representations of numbers that aren't exact, for example $\sqrt{2}\approx 1.41$. Such representations may be obtained using differentials (more generally, Taylor's formula), linear interpolation, etc.

4607 questions
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Using Stirling approximation find $\lim_{n\to \infty} \frac {n!}{e^n} $

What is $\lim_{n\to \infty}\frac {n!}{e^n} ? $ e.g the expression $\frac {n!}{e^n}$ approximates to what as n gets larger? Here I should use Stirling approximation which is $n!\approx \sqrt {2\pi n}(\frac {n}{e})^n$ How to approach now? Any…
Leyla Alkan
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How can I approximate this equation?

The approximation I'm having trouble with is this $$V = a\ln\left(\frac{1+\frac{L}{a}}{1-\frac{L}{a}}\right)-2L,\space (a\gg L)$$ The hint was to use $\sqrt{1+x^2}=1+\frac{1}{2}x^2+\,...$ and $\log(1+x)=x-\frac{1}{2}x^2+\,...$ I couldn't find a way…
Zack D
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How to find the positive solution to $\frac{x^d}{d} = 2^x$?

I asked Wolfram, and it gave me a complicated answer involving the ProductLog function $W$. Is there a simpler, approximate answer?
Erel Segal-Halevi
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Approximation algorithm for sine function

What is the most common approximation algorithm for sine function which can be implemented with self-organizing maps algorithm? I was thinking of least-squares approximation but I can't find a description of it for sine(or any trigonometrical)…
Sergey
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If value halves every $5$ years, when will the dollar be worth $1/1,000,000$ its current value?

This was a GRE multiple choice question. At a $15$ percent annual inflation rate, the dollar would decrease by approximately one-half every $5$ years. At this inflation rate, in approximately how many years would the dollar be worth…
McGuiness
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Approximate value of k

How do you solve $k$ in $\frac{(k-1)^{k-1}}{k^{k-2}}=n$ at least with a good approximation? Is there tight approximation?
Turbo
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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.
Berziks
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Trying to rederive an exponential approximation

So I was reading a paper where the following approximation was made. Note that $p$ is small, $L$ is large, and $pL$ is $O(1)$: $$\left[1-e^{-4p(1+p)L}\right]^{L/2}=\textrm{exp}\left[\frac{2(pL)^2}{e^{4pL}-1}\right](1-e^{-4pL})^{L/2}+O(p^3L^2)$$ I…
user34364
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Queries regarding saddle point

I was reading this article in wikipedia related to saddle points. When I came across this line In one dimension, a saddle point is a point which is both a stationary point and a point of inflection. Since it is a point of inflection, it is not a…
rajan sthapit
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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 approximation is done $f(x) \approx x + cx^2(1-x)$ So how is…
nist
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How to choose the extent of approximation for this equation?

Given the curve $$y={1 \over{x^2-1}}-{1\over {x^2}}$$ Find reasonable approximations for the intersections of this curve with the straight line $y=Ax$ (a) when $A$ is a very small positive number (b) when $A$ is a very large positive number. This…
Rescy_
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Get the number of digits $n$ is accurate to $p$

I am writing a little something on the accuracy of of approximations of certain numbers. Currently, I'm looking for a way to find the number of digits a number $n$ (the approximation) is "good for" $p$ (the number to be approximated). For example,…
Conor O'Brien
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How many significant figures are needed in base 2?

$x \in \mathbb{R}$ $2^{500}
Must
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Assymptotically approximating a sum similar to binomial.

I am, through some combinatorial problems which I'm working on, trying to figure out what the following sum becomes as $n\rightarrow \infty$: \begin{equation*} \sum_{i=1}^{n-1}…
Ove Ahlman
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Multivariate B-Spline Derivatives

To construct a multivariate B-spline, we simply take the Kronecker tensor product between the univariate basis functions constructed for each individual dimension. What I'd like to know is how do you construct a B-spline derivative for a…
Projectile Fish
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