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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Approximation of a generalized harmonic series

Take a sequence of positive integers $\{n_d\}_{d\in\Bbb N}$ and fix two positive real numbers $x$ and $y$. Is there a way to estimate $$ \sum_{d=1}^N\frac{n_d}{x+dy} $$ in function of $x,y$ and $N$?
Joe
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Please check my approximation . $~\frac{a+b}{b-a}=\frac{2a+b-a}{b-a}\approx\frac{2a}{b-a}$

$$ \left( a,b \in \mathbb R_{> 0} \right) ~~\wedge~~ \left( a < b\right) ~~\wedge~~ \left( \left( b-a \right) \ll a \right) $$ I want to derive the below appoximation equation . $$ \frac{ a+b }{ b-a } =\frac{ 2a+b-a }{ b-a } \approx \frac{…
electrical apprentice
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How fast can $\sin(ax)$ be uniformly approximated by polynomials on $[-1,1]$?

$a$ is a real number. Let $B_n(x)$ be the best uniform approximation degree-$n$ polynomial of $\sin(ax)$ on $[-1,1]$. Denote $R_n = ||B_n(x)-\sin(ax)||_{L^{\infty}}$. Does there exist some function $u(n), v(n)$ such that $$O(u(n))\le R_n\le…
qdmj
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Understanding Newton's approximation method with an example

Let $f(x)=x^2+3x-10$. If we take $x_0=0$ as the starting point for the Newton method, then the value of $x_2$ is what? I have tried but it appears I am wrong somewhere as the exercise from where I have taken this problem does not matches my answer…
Splendid Digital Solutions
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Does the commutative property of addition holds for uncertainties?

Box A has a height of $2.0±0.1$m, box B has a height of $1.5±0.1$m. If box B is placed on top of box A, what is the total height of both boxes? I know that to find the total height of both boxes, I have to do like the following $$A_{height} = 2.0…
Mohammad muazzam ali
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two-term asymptotic expansion for each root of the polynomial equation

two-term asymptotic expansion for each root: $\epsilon z^8-z^3-1=0$ as $\epsilon \rightarrow 0$ how to find other solutions than the one near -1? first, I check when $\epsilon=0$, I get the root of $z=-1$, though I am not sure if this root is of…
zhizhi
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Can you use an equals sign if the answer is rounded to two decimal places?

For my latest maths PSMT, the teacher has specified that equals signs should only be used if the values are actually equal. Does that mean I should use an approximation sign if I'm rounding the values to two decimal places? This is what I've been…
blackjack_sparrow
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How to find most efficient approximation of a real number with combination of positive and negative powers of two?

I need to approximate a real number $x$ as a sum of some set of positive and negative powers of two, e.g. $$ x = \delta_0 2^{n_0} + \delta_1 2^{n_1} + \delta_2 2^{n_2} + \ldots $$ where each $\delta_i$ is $-1$ or $+1$ and each $n_i$ is some integer.…
aquaticapetheory
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Deducing the approximation using double greater-than

I've been struggling against the below deduction of the approximation. $0
user802763
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Please help with this approximation problem

Question : If you approximate $2.7145$ to $2.715$ then write the percentage relative error. A) $0.000106$% B)$0.00106$% C) $0.0106$% D) $0.106$% I tried this by using the following approach : $$100 \times\frac {\text{assumed value }- \text{ original…
Prachi jain
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Approximation to simplify an equation

I found this approximation on the solution of a book. $$\left(1+\frac{\Delta^2}{d^2}\right)^{\frac {-3} {2} }+\left(1-\frac{\Delta^2}{d^2}\right)^{-2}= \left(1-\frac{3}{2}\frac{\Delta^2}{d^2}+...\right)+\left(1+\frac{2\Delta^2}{d^2}+...\right)$$ Can…
SharonZh
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Distribution of square numbers in intervale

The square numbers are of the form $n^{2}$ $(1,4,9,16,...)$ My question is there some formula to know how many square numbers up to $x$? or a least approximation formula ?
Abdo
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Approximation for the second order derivative: $\frac{d^2y}{dx^2}\approx\frac{\Delta y}{\Delta x^2}$. Why?

My physics professor made the following approximation: $$\frac{d^2y}{dx^2}\approx\frac{\Delta y}{\Delta x^2}$$ How can you fundament this? I get how you can do $\displaystyle\frac{dy}{dx}\approx\frac{\Delta y}{\Delta x}$, but it doesn't really click…
Belen
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Can't understand this approximation

This seems like it should be simple but I can't understand it. I'm reading this paper about spiral structure in galaxies. It says "this local shrinkage has increased the density, μ, per unit surface area in that neighbourhood by a fraction ε ... so…
maurice
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Mollifier at the discontinuous point

Let $f\in C^0(S^1\backslash\{p\})\cap L^\infty(S^1)$ for some $p\in S^1$. I guess $$\lim_{\epsilon\rightarrow 0}f_\epsilon(p)=\frac{1}{2}\lim_{x\rightarrow p-}f(x)+\frac{1}{2}\lim_{x\rightarrow p+}f(x)$$ when $f$ is nice enough, where $f_\epsilon$…
Jingeon An-Lacroix
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