# Tagged Questions

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### Rational approximation of $\tanh\,(\sqrt[4]{s}$)

I'd like to find a rational representation of $$f(s) = \frac{\tanh\,\sqrt[4]{s}}{\sqrt[4]{s}}= \frac{a_0 + a_1 s + a_2 s^2 + ... + a_n s^n}{b_0 + b_1 s + b_2 s^2 + ... + b_m s^m}$$ For the case ...
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### Very slow convergence of a particular series?

I've read that $$\sum_{k=2}^{\infty} \frac{1}{k (\log k)^2} = 2.1097\ldots$$ However when I compute the partial sums it looks like a lot of terms are needed to even get the first decimals right. My ...
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### Fastest Square Root Algorithm

What is the fastest algorithm for finding the square root of a number? I created one that can find the square root of "987654321" to 16 decimal places in just 20 iterations (I'm not ready to release ...
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### Accelerating Convergence of a Sequence

Suppose I had a monotonically increasing sequence $\{d_{n}\}$ which is also bounded above. The $d_{n}$'s satisfy a given recurrence, however computationally they tend very slowly to the limit. What ...
### Calculate $\pi$ in an arbitrary base, to arbitrary precision
I need to calculate $\pi$ -- in base: 4, 12, 32, and 128 -- to an arbitrary number of digits. (It's for an artist friend). I remember Taylor series and I've found miscellaneous "BBP" formulas, but so ...
Notation: $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right)$ is the Gauss Notation for generalized continued fractions. Description of the Bauer-Muir transformation ...