Closed form of a sum $ \sum_{i=1}^{n}\frac{1}{((i-1)x)^2+y^2}$ Consider a sum:
$$ \sum_{i=1}^{n}\frac{1}{((i-1)x)^2+y^2}$$
with $x$ and $y$ being (non-zero) constants. Is it possible to obtain a nice closed form of this expression?

------ Complementing Jack's answer  -----
Actually, since we have the formula for the infinite summation
$$
S\left( {x,y} \right) = \sum\limits_{0\, \leqslant \,k} {\;\frac{1}
{{\left( {x + k} \right)^{\,2}  + y^{\,2} }}}  = \frac{1}
{{2y^{\,2} }} + \frac{{\pi \coth \left( {\pi y/x} \right)}}
{{2xy}}
$$
then, for a partial one we will have
$$
\begin{gathered}
  S\left( {x,y,n} \right) = \sum\limits_{0\, \leqslant \,k\, \leqslant \,n - 1} {\;\frac{1}
{{\left( {x + k} \right)^{\,2}  + y^{\,2} }}}  =  \hfill \\
   = \sum\limits_{0\, \leqslant \,k} {\;\frac{1}
{{\left( {x + k} \right)^{\,2}  + y^{\,2} }}}  - \sum\limits_{n\, \leqslant \,k} {\;\frac{1}
{{\left( {x + k} \right)^{\,2}  + y^{\,2} }}}  =  \hfill \\
   = \sum\limits_{0\, \leqslant \,k} {\;\frac{1}
{{\left( {x + k} \right)^{\,2}  + y^{\,2} }}}  - \sum\limits_{0\, \leqslant \,k} {\;\frac{1}
{{\left( {\left( {x + n} \right) + k} \right)^{\,2}  + y^{\,2} }}}  =  \hfill \\
   = S\left( {x,y} \right) - S\left( {x + n,y} \right) \hfill \\ 
\end{gathered} 
$$  
------ in another way  ----- 
$$
\begin{gathered}
  S\left( {x,y,n} \right) = \sum\limits_{0\, \leqslant \,k\, \leqslant \,n - 1} {\;\frac{1}
{{\left( {x + k} \right)^{\,2}  + y^{\,2} }}}  = \sum\limits_{0\, \leqslant \,k\, \leqslant \,n - 1} {\;\frac{1}
{{\left( {x + k} \right)^{\,2}  - \left( {iy} \right)^{\,2} }}}  =  \hfill \\
   = \sum\limits_{0\, \leqslant \,k\, \leqslant \,n - 1} {\;\frac{1}
{{\left( {x + iy + k} \right)\left( {x - iy + k} \right)}}}  = \frac{1}
{{2iy}}\left( {\sum\limits_{0\, \leqslant \,k\, \leqslant \,n - 1} {\;\frac{1}
{{\left( {x - iy + k} \right)}}}  - \sum\limits_{0\, \leqslant \,k} {\;\frac{1}
{{\left( {x + iy + k} \right)}}} } \right) =  \hfill \\
   = \frac{1}
{{2iy}}\left( {\psi ^{\left( 0 \right)} \left( {x - iy + n} \right) - \psi ^{\left( 0 \right)} \left( {x + iy + n} \right) + \psi ^{\left( 0 \right)} \left( {x + iy} \right) - \psi ^{\left( 0 \right)} \left( {x - iy} \right)} \right) =  \hfill \\
   = \frac{1}
{y}\left( {\operatorname{Im} \left( {\psi ^{\left( 0 \right)} \left( {x + iy} \right)} \right) - \operatorname{Im} \left( {\psi ^{\left( 0 \right)} \left( {x + iy + n} \right)} \right)} \right) \hfill \\ 
\end{gathered} 
$$
