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### Guaranteed correct digits of elementary expressions

There is a certain confusion in the answers, so let me try to dispel this confusion. There are two different issues here. One is “computing an approximation with arbitrary precision” and one is “...
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### Guaranteed correct digits of elementary expressions

This problem is Turing equivalent to the constant problem (see also Wikipedia-constant problem), but it is open whether this problem is decidable. The constant problem is the problem of determining ...

### Are these fast convergent series for $\log(2)$, $\log(3)$ and $\log(5)$ already known and proven?

By naming $S_2,S_1$ and $S_3$ the rhs series in Eqs.(1), (3) and (4) respectively, a proof that $S_\ell=\log\left(\frac{7+\ell}{5-\ell}\right)$ for $\ell=1,2,3$ is built transforming them to an ...
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### Are these fast convergent series for $\log(2)$, $\log(3)$ and $\log(5)$ already known and proven?

A few superficial low-tech remarks that probably aren't useful at all, just in case they spark more useful ideas for others. It's easy to rewrite the "quotient of product of rising factorials&...

### Guaranteed correct digits of elementary expressions

Edited after the comment of Joel David Hamkins. The "correct $n$-th digit of a real number, of of real and imaginary part of a complex number is ill defined. For example, what is the $n$-th digit ...
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### Are these fast convergent series for $\log(2)$, $\log(3)$ and $\log(5)$ already known and proven?
As a supplement of @Jorge Zuniga's answer, we can obtain more similar series which are $\pi$ and $\log 2$ related via the same idea, such as:  \begin{eqnarray} \log2&=&\sum_{n=0}^{\infty}\...
The answer is yes. Indeed, the cdf $F$ of the binomial distribution is log concave -- see e.g. Theorem 2 on p. 152, used with $\alpha=1$, $r=\infty$, and $q$ being the probability mass function of the ...