A reciprocal area and the endpoint choice

Example 2 (Self-Reading). The graph $y=1/x$ on $[1,2]$ has $n$ rectangles of equal width, using right endpoints. Show that their total area is $S_n=\sum_{r=1}^n\frac1{n+r}$ and find the exact value of $\lim_{n\to\infty}S_n$. Teacher check: construct a similar diagram using left endpoints.