Notice that $\frac{2^{2}+1}{2^{2}-1}+\frac{3^{2}+1}{3^{2}-1}+...+\frac{2010^{2}+1}{2010^{2}-1}=2009+2(\frac{1}{2^2-1}+\frac{1}{3^2-1}+...+\frac{1}{2010^2-1})$ $=2009+2(\frac{1}{(1)(3)}+\frac{1}{(2)(4)}+...+\frac{1}{(2009)(2011)})$ $=2009+(\frac{1}{1}-\frac{1}{3})+(\frac{1}{2}-\frac{1}{4})...+(\frac{1}{2009}-\frac{1}{2011})$ $=2009+1+\frac{1}{2}-\frac{1}{2010}-\frac{1}{2011}$ $2010<2010+\frac{1}{2}-\frac{1}{2010}-\frac{1}{2011}<2010+\frac{1}{2}$

$\sum_{n=a}^{k} \frac{n^2+1}{n^2-1} = \sum_{n=a}^{k} a+\frac{1}{(n+1)(n-1)} = k-a+1+\sum_{n=a}{k} \frac{1}{n-1}-\frac{1}{n+1} =$ $\boxed{ k-a+1+\frac{1}{a-1}+\frac{1}a-\frac{1}{k}-\frac{1}{k+1}}$