User:Jaume Oliver Lafont/Generalized Mercator series

1 $log(n)=\sum _{k=1}^{\infty }\left({\frac {1}{k}}-{\frac {1}{n\lceil {\frac {k}{n}}\rceil }}\right)$

2 $log(p)=-\sum _{n=1}^{\infty }{\frac {1}{n}}\sum _{k=1}^{p-1}e^{2\pi i{\frac {kn}{p}}}$

A167407

3 $log(p)=\lim _{n\to \infty }\sum _{k=n+1}^{np}{\frac {1}{k}}$

(Eric Naslund answered

$log(n)=...=\lim _{M\to \infty }\sum _{k=1}^{nM}{\frac {1}{k}}-\sum _{k=1}^{M}{\frac {1}{k}}$

to this question by Mats Granvik)