Approximate ln(2) out of small numbers

Question

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Given numbers 1,2,...n, the goal of this puzzle is to make a number as close to the mathematical constant $\ln(2) ≈ 0.69314718055$ as possible.

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Rules

• You can only use the four mathematical operations $+,-,\times,/$ and parentheses/brackets $()[]$.
• It is not required to use all of the numbers.
• You cannot concatenate (stick) the numbers together. (i.e. you can't get $12$ from $1,2$)

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Examples:

$n$	Approximation	Error
1	1	$3.068\cdot10^{-1}$
2	$\frac12=0.5$	$1.931\cdot10^{-1}$
3	$\frac23=0.\overline6$	$2.6481\cdot10^{-2}$
7	$\frac7{6+4+(1/(3\times2+5))}=\frac{77}{111}=0.\overline{693}$	$5.4651\cdot10^{-4}$
8	$\frac7{6+4+(2/(3\times8+1-5))}=\frac{70}{101}=0.\overline{6930}$	$7.7873\cdot10^{-5}$

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Give your best approximation for n=1, 2, 3...20. (Give a possibly different answer for each n)

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If anyone is curious I am currently only aware of optimal solutions for $n=1,2...8$.

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Answer 1

An approach (hinted at by @michaelmoschella in the comments) is to use the fraction approximation algorithm [insert citation].

Something like this:

Here we start with 0/1, 1/1, and 1/0 ($=\infty$). Then we look which side our target number is in (left or right). Those numbers become the new boundaries, and the new middle numerator (denominator) is the sum of the new outer numerators (denominators).

I've highlighted the "best" rows, which tend to be the last best so far (so when it's the best so far, but the next row is not).

See here for the calcs.

So, for example, going after 445/642 is a reasonably fruitful exercise (7.63E-7).

$445 = (9\times10-1)\times5$

$642 = (11\times10-3)\times6$

Oh, but whoops, I've repeated the 10. But I could replace that we $(8+2)$.

So I get ($n=11$): $\frac{(9\times10-1)\times5}{(11\times(8+2)-3)\times6}$

I doubt it's the best, but it's a start.

Of course, the above is great for fractions, but we're here allowed to do more than that, so there may also be better approaches altogether.

You can also see $\frac{4+3}{2*5}$ for $n=5$ has an error of 6.85E-3.