How to solve $\sum_{k=1}^n\frac{k}{n^2+k}$? Can someone show me what is wrong with the expression I got for evaluating $\sum_{k=1}^n\frac{k}{n^2+k}$?
Steps:  
$\sum_{k=1}^n\frac{k}{n^2+k} = \frac{\sum_{k=1}^nk}{\sum_{k=1}^{n}n^2+k} =  \frac{\frac{1}{2}n(n+1)}{\sum_{k=1}^{n}n^2+\sum_{k=1}^{n}k} = \frac{\frac{1}{2}n(n+1)}{n^3+\frac{1}{2}n(n+1)} = \frac{n(n+1)}{2n^3+n(n+1)} = \frac{n+1}{2n^2+n+1}$  
However, when I plug in something like $\sum_{k=1}^{10}\frac{k}{10^2+k}$ into this online summation calculator, the answer I get is $\sim0.514$, which is not the same answer I get with $\frac{10+1}{2(10)^2+10+1} = \frac{11}{211} \approx 0.0521$

Your first step is wrong:
$$\sum_{k=1}^n \frac{k}{n^2+k} \neq \frac{\sum_{k=1}^n k}{\sum_{k=1}^n n^2+k}$$
It's like saying $\frac ab + \frac cd=\frac{a+c}{b+d}$ which is false...
