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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The approximation formula $\left|\alpha -\frac{p}{q}\right| \le \frac{1}{\sqrt{5}q^2}$

I have seen a result about the approximation of irrational numbers and want to find its proof. Suppose $\alpha$ is an irrational number, then there are infinitely many integers $p,q$ with $(p,q)=1$ such that $$\left|\alpha -\frac{p}{q}\right|…
No One
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Proof that Newton's Method gives better and better approximations with each iteration?

I've seen this question and answer: Why does Newton's method work? It gives some geometric intuition as to what is going on when applying Newton's method, but what I really need to know is why it works that way. How can you prove that each…
jeremy radcliff
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Ideas on how to simplify or approximate this nasty sum

I have a sum (let's call it $p$): $$p:= \frac{1}{n!}\sum_{i=2}^l (i-1)\frac{(n-k)!}{(n-k-i+2)!}(n-i)!$$ where $l, n, k$ are fixed positive integers, and $k \leq n$. I'd like to either simplify or approximate $p$, for the purpose of simplifying or…
user341502
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Approximation of $\sqrt{2}$

I got the following problem in a chapter of approximations: If $\frac{m}{n}$ is an approximation to $\sqrt{2}$ then prove that $\frac{m}{2n}+\frac{n}{m}$ is a better approximation to $\sqrt{2}.$(where $\frac{m}{n}$ is a rational number) I am stuck…
user333900
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What's about $\sum_{n=1}^{\infty} \frac{e^{H_n}\log H_n}{n^3}$, where $H_n$ is the nth harmonic number?

I would like to do a toy verification of the Riemann hypothesis exploiting theLagarias theorem (see the section Applications in the following link) and the fact that we know a lot of decimals for Apèry's constant. Combining Lagarias, with the…
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Probability of alternating sequence from uniform distribution

Say I sample a discrete uniform distribution $U$ (say $U$ has a support of $N$ elements, and there is a total order on the elements) a number $K$ times, resulting in a random sequence $A_{i}$. What can I say about the probability that $A_{i}$…
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Uniform approximation. Two boundary layers?

Find uniform approximation up to order $O(\epsilon)$: $$ \begin{cases} \epsilon y''+\epsilon y' - y^2=-1-x^2 \\ y(0)=2 \\ y(1)=2 \end{cases} $$ At $\epsilon=0$ solutions $\pm \sqrt{1+x^2}$ don't satisfy either of the B.C. Does this mean there are…
user16015
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Approximating Fresnel integrals with standard functions

I would like to approximate the Fresnel S and Fresnel C with standard functions. I've started with the $ S(x) $ function: $$ approxS(x) = sgn(x) * \left ( sgn(x)* \left ( \frac{ \sin( \frac{x^2}{2} ) }{x} \right) + 0.5 \right ) $$ The result looks…
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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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Polynomial Approximation of an Integral

I require a polynomial $p(x)$ such that $$\left|p(x) - \int_0^x \cos{(t^2)} dt\right| < \frac{1}{10!}$$ for all $x \in [-1, 1]$. I know that I should probably use the fact that if $$m\leq f^{n+1} {(t)} \leq M$$ for $t$ in an interval containing the…
user36387
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Upper bound on ratio of incomplete Gamma function and Gamma function $\frac{ \Gamma \left( x; a\right)}{\Gamma(x)}$

I am trying to find a tight upper bound the following expression \begin{align} \frac{ \Gamma \left( x; a\right)}{\Gamma(x)} \end{align} where $\Gamma \left( x; a\right)$ is incomplet Gamma function \begin{align} \Gamma \left( x; a\right)=…
Boby
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Approximating number of partitions of $n$, denoted $p(n)$, by $p(n)\ge e^{c\sqrt n}$

I was to show that $p(n)$, the number of partition of a positive integer $n$, satisfy: $p(n)\ge \max_{1\le k\le n}{{n-1\choose k-1}\over k!}$, which was obvious because every unordered collection of numbers has at most $k!$ ordered…
Meitar
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Approximate solution for the root of a non-linear function

I have been working with a system which involves computing the roots of functions that look like \begin{equation} e^t (g\cos(\omega t) + b) = c \end{equation} where $t$ is the independent variable and the parameters $g$, $\omega$ and $b$ are real…
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Approximating $f(1)$ given $f(0)=2$ and $\frac{1}{2}x^2≤f'(x)≤x^2$?

This is an exercise using the mean value theorem: Approximate $f(1)$ given $f(0)=2$ and $\frac{1}{2}x^2≤f'(x)≤x^2$ for $x≥0$. I've found (using MVT): $$\frac{1}{2}x^2+2≤f(1)≤x^2+2$$ and I can constrain this to $$2≤f(1)≤x^2+2$$ by noting that $x≥0$…
mavavilj
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How to show that $2 ^\binom{N}{2} \sim \exp(\frac{N^2}{2}\ln(2))$

How can we show that: $2 ^\binom{N}{2} \sim \exp(\frac{N^2}{2}\ln(2))$ for large N.
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