# 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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### A 1,400 years old approximation to the sine function by Mahabhaskariya of Bhaskara I

The approximation $$\sin(x) \simeq \frac{16 (\pi -x) x}{5 \pi ^2-4 (\pi -x) x}\qquad (0\leq x\leq\pi)$$ was proposed by Mahabhaskariya of Bhaskara I, a seventh-century Indian mathematician. I ...
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### Simple numerical methods for calculating the digits of $\pi$

Are there any simple methods for calculating the digits of $\pi$? Computers are able to calculate billions of digits, so there must be an algorithm for computing them. Is there a simple algorithm that ...
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### Approximation of $e$ using $\pi$ and $\phi$?

$$e \approx \frac{4 \phi +3 \pi-5}{4}$$ where $~\phi~$ is a Golden ratio . Is it possible to construct better approximation of $e$ using $\pi$ , $\phi$ and integers ?
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### $\lim_{n\to\infty} f(2^n)$ for some very slowly increasing function $f(n)$

I should be able to answer this myself, but feel insecure anyway. I want to know, whether a function f(n) is bounded if n goes to infinity (and if it's bounded, the limit). Heuristically it appears (...
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### Why is $\pi^2$ so close to $10$?

Noam Elkies explained why $\pi^2=9.8696...$ is so close to $10$ using an inequality on Euler's solution to Basel problem $$\frac{\pi^2}{6}=\sum_{k=0}^{\infty} \frac{1}{\left(k+1\right)^2}$$ to form ...
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### Why $e^{\pi}-\pi \approx 20$, and $e^{2\pi}-24 \approx 2^9$?

This was inspired by this post. Let $q = e^{2\pi\,i\tau}$. Then, $$x := \left(\frac{\eta(\tau)}{\eta(2\tau)}\right)^{24} = \frac{1}{q} - 24 + 276q - 2048q^2 + 11202q^3 - 49152q^4+ \cdots\tag1$$ ...
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### Approximating $\pi$ with least digits

Do you a digit efficient way to approximate $\pi$? I mean representing many digits of $\pi$ using only a few numeric digits and some sort of equation. Maybe mathematical operations also count as ...
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### Series for Stieltjes constants from $\gamma= \sum_{n=1}^\infty \left(\frac{2}{n}-\sum_{j=n(n-1)+1}^{n(n+1)} \frac{1}{j}\right)$

Euler's constant has the following representations (Euler-Mascheroni constant expression, further simplification, http://math.stackexchange.com/a/129808/134791, Question on Macys formula for Euler-...
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### How to calculate square root or cube root? [duplicate]

I was reading Richard Feynman biography when I read that one time he was able to calculate the cube root of large number in his brain by just using simple facts of everyday life. So my question is ...
### Method for estimating the $n^{th}$ derivative?
When using numerical analysis, I often find that I am required to estimate a derivative (e.g. when using Newton Iteration for finding roots). To estimate the first derivative of a function $f(x)$ at ...
### Approximating $\arctan x$ for large $|x|$
I would like to know if there is reasonably fast converging method for computing large arguments of arctan. Until now I've came across Taylor series that converges only on interval $(-1,1)$ and for ...