Logging The Summation!!

A Quick Solution: $\dfrac { (r-n)({ r }^{ 2 }+{ n }^{ 2 }+nr) }{ (\log { r } -\log { n){ n }^{ 4 } } } =\dfrac { { r }^{ 3 }-{ n }^{ 3 } }{ { n }^{ 4 }\log { (\frac { r }{ n } ) } }$ $=\dfrac { \frac { { r }^{ 3 } }{ { n }^{ 3 } } -1 }{ \log { \frac { r }{ n } } } \frac { 1 }{ n }$

Now using Riemann Sums: $=$ $\displaystyle \int _{ 0 }^{ 1 }{ \frac { { x }^{ 3 }-1 }{ \log { x } } dx\quad }$

This is a standard integral which can be solved by differentiating through the integral, the answer comes out to be: $\ln(4)$ According to the condition given the answer to be submitted is: $\log { 4 } \approx 0.6020$