Problem - 4561
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.
Yes, this limit exists. Note that
$$S_n=\sum_{k=1}^{2n}\frac{1}{n+k}=\sum_{k=1}^{2n}=\sum_{k=1}^{2n}\frac{1}{n}\cdot\frac{1}{1+\frac{k}{n}}$$
is the Riemann sum of
$$\int_0^2\frac{1}{1+x}dx = \left.\ln{(1+x)}\right|_0^2=\boxed{\ln {3}}$$