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.

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Approximating $e^{x}/(e^{x} - 1)$

Is it correct to tell that we can approximate \begin{equation*} \frac{e^{x}}{(e^{x} - 1)} \end{equation*} by: \begin{equation*} \frac{1}{x} \end{equation*}
watou
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Approximation of an expression (no calculator please!)

Today I had my college admission exam, It was good, but there was a question which I found a bit interesting (but unable to solve at the moment). It says, Question: Find the positive integer which is just equal to the expression …
Saharsh
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Why are these sums approximately equal?

Let $T$ be a finite set. Let $\rho:T\rightarrow (0,1)$ be such that $\sum_{t\in T}\rho(t)=1$. Let $F:\mathbb N\cup\{0\}\rightarrow(0,1)$ be such that $\sum_{i=0}^\infty F(i)=1$. Let $\mu_F=\sum_{i=0}^\infty iF(i)$. Let $\ell:T\rightarrow\mathbb N$…
Gregory Grant
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if $x\ll 1$ is it safe to assume that $x\ll \frac{1}{2}$

I know that: if $x\ll 1$ then we can write $\frac{x}{x+1}\rightarrow x$ but is it safe to write $\frac{2x+1}{x+1}\rightarrow 1$?
asdfgg
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Approximation: is it logical to approximate to zero?

"What is the value of 0.02 cm rounded to the nearest centimeter?" Is it logical to approximate a real value (however small) to zero? I know that following a simple 'rounding' or approximation algorithm, the answer is zero cm. This offends my sense…
cormac
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Approximating $a = bx + cx^2 + O(x^3)$

I'm attempting to work out a problem of the form $a = bx + cx^2 + O(x^3)$, where I need to solve for $x$. To be honest, I don't really know how to work this out. Someone suggested that it can be done recursively, by writing $x = \frac{a}{b}…
user3183724
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What are some quickly convergent, easily calculated approximations for common functions for when you've forgotten a calculator to a test?

I think it's nice not to rely too much on a calculator, whether it's forgotten or forbidden. Approximations can be useful on exams when you want a good guess at the answer to see if it's somewhat correct. Evaluating decimals are often better than…
Frank Vel
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How is this approximation related?

I'm in the dark about the following approximation. Given that $r >> (l_1 + l_2)$ $\phi(P) \approx \dfrac{\lambda}{4\pi\epsilon_0}ln(\dfrac{r+l_2}{r-l_1}) \approx \dfrac{\lambda(l_1 + l_2)}{4\pi\epsilon_0 r}$ I do not see the relation. If possible…
user3346148
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Approximaton to sin (more precisely)

same story, If we have $s$ circle with the diameter $AB$ (with length $1$) and the center $O$, then we can approximate $\operatorname{chord} AC$ where $x$ represents the value of the $\angle AOC$ in degrees, and $t=90-\frac{x}{2}$.So formula…
Srbin
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Understanding Newton iterative steps with an example

For the function $f(x)=x^2-a, a>0,$ the Newton iterative steps are given by? $$ x_{n+1} = \frac 12 [x_n + \frac {a}{x_n}]$$ $$ x_{n+1} = 2[x_n + \frac {a}{x_n}]$$ $$ x_{n+1} = \frac 12 [x_n -ax_n]$$ $$ x_{n+1} = \frac 12 [x_n + ax_n]$$ Clueless…
Splendid Digital Solutions
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How would I go about proving this approximation?

I was just playing around in desmos and I think I found something that approximates Lambert W for whole numbers. $$f(x)=a(\int_{0}^{1}(\sum_{n=1}^{x}t^n-1)dt)+b$$ Where $a\approx0.765424$ And $b\approx0.944602$ $a$ and $b$ were achieved through…
CoderBoy
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Determine the (2,2) Pade approximation to sqrt cube to x+8 and estimate its error

here is the full question its about pade approximation and I serch the internet and couldn't find the answer anyone can help it will be great
John wallas
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Numerical Analysis: Approximations -- Discrete Average Value Theorem

I am asked to compute approximations to $f'(1)$ using $h=\frac{1}{16}$ for $f(x)=\sqrt{x+1}$ with the following formulas $$f'(x)-\frac{f(x+h)-f(x)}{h}=-\frac{1}{2}hf''(\xi_{x,h})=\mathcal{O}(h)....(1)$$…
cryptomath
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How to arrive at this approximation?

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{\delta{z_{n}} -\delta{z_{n-j}}}{(ja)^3} -…
aswa09
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How to get the approximation of $\ln 2$?

How to get the approximation of $\ln 2$ and prove results using the knowledge of senior high school(China)See in the textbook.(the figure accurate to the third decimal place)? Above the question,everything we are supposed to use(without…
icase233
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