What makes you think the limit is 0? For any n, this is the sum of n terms, each of which is positive! How could the limit, as n goes to infinity, be 0?
As a "Riemann sum" we have an interval of length 4, divided into n sub-intervals, so that each sub-interval has length 4/n. We are summing a function of the form \(\displaystyle \frac{1}{3+ \frac{4i/n}}\) with i going from 0 to n. In the limit as n goes to infinity that becomes the integral \(\displaystyle \int_0^4 \frac{1}{3+ 4x}dx\(\displaystyle . Let u= 3+ 4x so that when x= 0, u= 3 and when x= 4,u= 7. du= 4dx so dx= du/4. The integral becomes \(\displaystyle \frac{1}{4}\int_4^7 \frac{1}{u}= \left[ln(|u|)\right]_4^7= ln(7)- ln(4)= ln\left(frac{7}{4}\right)\).
The limit, as n goes to infinity, of this expression, is \(\displaystyle ln\left(frac{7}{4}\right)\).\)\)
