# Tagged Questions

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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### Are there known pairs of simple numbers equal to huge precision, but not equal strictly?

Are there known pairs of numbers $a$ and $b$, which at first look at them seemed likely to be equal, and after checking up to $10^n$ decimal places appeared to agree, but suddenly for some $n$ they ...
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### 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 ...
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### Taylor Formula: Lagrange's remainder vs Cauchy's remainder (and other less known forms)

While solving problems and exercises, so far I've only used Lagrange's form of the remainder. Indeed, it must be said that many textbooks don't even mention other forms of the remainder for Taylor's ...
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### Watson's Lemma Extension

We all know that Watson's Lemma is used to approximate the integral $$F\left( s \right)=\int_0^\infty {{e^{ - st}}f\left( t \right)dt}$$ for large $s$. However, for arbitrary $s$, are there any ...
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### How can I use Stirlings inequality to prove this inequality?

Let $p,k$ be natural numbers with $p\ge k$, show that $$\frac{(p-k)!}{(p+k)!} \le \frac{1}{p^{2k}} \left(\frac{e}{2} \right)^{2k^2/p}.$$ The text where I come across this says to use Stirling's ...
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### Shaking a box of rocks (Optimal Packing)

My coworker was telling me that when he plants seeds on his farm, he puts them all in a large container on the tractor and after a period of just driving, the seeds are more densely packed than when ...
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### Asymptotic behavior of many derivatives

To compute the residue of a pole of very high order $M$ at $z=0$, one needs to compute $\frac{d^M}{dz^M} g(z)$ Suppose that $g(z)$ is a reasonable but not trivial function, that itself may depend on ...
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### Approximate the integral $\int_0^\pi \sin(x^3)\mathrm{d}x$ with a standard pocket calculator

I came over the following integral $$\int_0^\pi \sin(x^3) \mathrm{d}x$$ when a friend of mine tried to approximate it. The most obvious way is to use taylors formula, and then turn the integral ...
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### Using formal power series to solve nasty equations

Consider a function $f:[0,\infty)\times \mathbb R\to\mathbb R$, and suppose that given some $a>0$, I would like to solve for $x\in\mathbb R$ satisfying \begin{align} f(\delta, x) = a. \end{align} ...
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### Approximating this definite integral

I ran into the following integral in my research that I believe has no closed-form solution: $$I = \int_{s_0}^{s_1} \frac{(\alpha_x s + \beta_x)^{\lambda_x}}{(\alpha_y s + \beta_y)^{\lambda_y}} ds$$...