Find the limit

Note that the limit can be re-written as From a Riemann Sum, we know that So,

Using the limit definition of an integral:

$\displaystyle\text{S}=\lim_{n\to\infty } { \quad \left( \frac { n }{ { \left( n+1 \right) }^{ 2 } } +\frac { n }{ { \left( n+2 \right) }^{ 2 } } +\frac { n }{ { \left( n+3 \right) }^{ 2 } } +...+\frac { 1 }{ 25n } \right)}=$

$\displaystyle\lim_{n\to\infty}\sum_{i=1}^{4n}\frac{n}{(n+i)^2}=\lim_{n\to\infty}\frac1{n}\sum_{i=1}^{4n}\frac{1}{(1+\frac{i}{n})^2}=\int_0^4\frac1{(1+x)^2}dx=\boxed{\frac4{5}}$