what is the limit of $\sum_{i=1}^n1/(2n+i)$ if $n$ goes to infinity?
Solution 1:
The summand equals $$\frac{1}{n}\cdot\frac{1}{2\frac{i}{n}+1}.$$
Can you interpret your sum as the Riemann sum of some function over the interval $[0,1]$? Hint: take an equipartition with $\Delta x_i=\frac{1}{n}$.