Also possibly of interest:

$$ \sum_{n=1}^\infty \left(-\frac{2}{b(n-1/b)(n+1/b)} + \frac{b}{n(n+1)}\right) = \pi \cot(\pi/b)$$

$$ \sum_{n=1}^\infty {\frac {t \left( {t}^{2}{n}^{2}+2\,{n}^{2}+2\,n+1 \right) }{ \left( n+ 1 \right) n \left( {t}^{2}{n}^{2}+1 \right) }} = \pi \coth(\pi/t) $$

EDIT: And, if I'm not mistaken,

$$ \sum_{n=1}^\infty \left(\frac{1-m}{mn} + \sum_{k=1}^{m-1} \frac{1}{mn-k}\right) = \ln(m) $$

Also possibly of interest:

$$ \sum_{n=1}^\infty \left(-\frac{2}{b(n-1/b)(n+1/b)} + \frac{b}{n(n+1)}\right) = \pi \cot(\pi/b)$$

$$ \sum_{n=1}^\infty {\frac {t \left( {t}^{2}{n}^{2}+2\,{n}^{2}+2\,n+1 \right) }{ \left( n+ 1 \right) n \left( {t}^{2}{n}^{2}+1 \right) }} = \pi \coth(\pi/t) $$

EDIT: And, if I'm not mistaken,

$$ \sum_{n=1}^\infty \left(\frac{1-m}{mn} + \sum_{k=1}^{m-1} \frac{1}{mn-k}\right) = \ln(m) $$ for positive integers $m$.

Also possibly of interest:

$$ \sum_{n=1}^\infty \left(-\frac{2}{b(n-1/b)(n+1/b)} + \frac{b}{n(n+1)}\right) = \pi \cot(\pi/b)$$

$$ \sum_{n=1}^\infty {\frac {t \left( {t}^{2}{n}^{2}+2\,{n}^{2}+2\,n+1 \right) }{ \left( n+ 1 \right) n \left( {t}^{2}{n}^{2}+1 \right) }} = \pi \coth(\pi/t) $$

Also possibly of interest:

$$ \sum_{n=1}^\infty \left(-\frac{2}{b(n-1/b)(n+1/b)} + \frac{b}{n(n+1)}\right) = \pi \cot(\pi/b)$$

$$ \sum_{n=1}^\infty {\frac {t \left( {t}^{2}{n}^{2}+2\,{n}^{2}+2\,n+1 \right) }{ \left( n+ 1 \right) n \left( {t}^{2}{n}^{2}+1 \right) }} = \pi \coth(\pi/t) $$

EDIT: And, if I'm not mistaken,

$$ \sum_{n=1}^\infty \left(\frac{1-m}{mn} + \sum_{k=1}^{m-1} \frac{1}{mn-k}\right) = \ln(m) $$