Let $$S_n=\sum_{k=1}^{2n}\frac{1}{n+k}=\frac{1}{n+1}+\frac{1}{n+2}+\cdots + \frac{1}{3n}$$

Does $\displaystyle\lim_{n\to\infty}S_n$ exist? If so, find its value. If not, prove the claim.