Motivation: In S.J. Patterson's An introduction to the theory of the Riemann Zeta-Function it is proved (p.132) that
$\displaystyle -\Gamma ^{\prime }(t)/\Gamma (t)=\gamma +t^{-1}+\underset{j\geq 1}{\sum }[(j+t)^{-1}-j^{-1}]$.
In Exercise A4.1 (p.135) one is asked to show that if $|t|<1$
$\displaystyle -\Gamma ^{\prime }(t)/\Gamma (t)=\gamma +t^{-1}+\underset{k\geq 1}{\sum }\zeta (k+1)(-t)^{k}$.
Question: How does one prove that
$\displaystyle\underset{j\geq 1}{\sum }[(j+t)^{-1}-j^{-1}]=\underset{k\geq 1}{\sum }\zeta (k+1)(-t)^{k}\qquad $for $|t|<1$?