Estonian Math Competitions

Answer: $\frac{505\cdot 1009}{2021}=\frac{509545}{2021}=252+\frac{253}{2021}$.

The sum consists of $1009$ terms, where the $i$-th term is of the form $\frac{i(i+2)}{(2i+1)(2i+3)}$. Let $s$ be the desired sum. Notice that $$\frac{i(i+2)}{(2i+1)(2i+3)}=\frac{1}{4}-\frac{3}{4}\cdot \frac{1}{(2i+1)(2i+3)}=\frac{1}{4}-\frac{3}{8}\cdot \left(\frac{1}{2i+1}-\frac{1}{2i+3}\right)$$ Therefore $$\begin{array}{rl}s& =1009\cdot \frac{1}{4}-\frac{3}{8}\cdot \left(\left(\frac{1}{3}-\frac{1}{5}\right)+\left(\frac{1}{5}-\frac{1}{7}\right)+\cdots +\left(\frac{1}{2019}-\frac{1}{2021}\right)\right)\\ & =\frac{1009}{4}-\frac{3}{8}\cdot \left(\frac{1}{3}-\frac{1}{2021}\right)=\frac{1009}{4}-\frac{3}{8}\cdot \frac{2021-3}{3\cdot 2021}=\frac{1009}{4}-\frac{2018}{8\cdot 2021}\\ & =\frac{1009}{4}-\frac{1009}{4\cdot 2021}=\frac{1009}{4}\cdot \left(1-\frac{1}{2021}\right)=\frac{1009}{4}\cdot \frac{2020}{2021}=\frac{505\cdot 1009}{2021}=\frac{509545}{2021}\end{array}$$