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Jaume Oliver Lafont

Electronics Engineer

The simplest rearrangements of the cancelling harmonic series converge to the logarithms of positive rationals.

$$ \log\left(\frac{p}{q}\right)=\sum_{i=0}^\infty \left(\sum_{j=pi+1}^{p(i+1)}\frac{1}{j}-\sum_{k=qi+1}^{q(i+1)}\frac{1}{k}\right) $$

$\pi^2$ is so close to $10$ because $$\sum_{k=0}^\infty\frac{1}{((k+1)(k+2))^3}$$ is small.

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