The limit of truncated sums of harmonic series, lim\lim\limits_{k\to\infty}\sum_{n=k+1}^{2k}{\frac{1}{n}}

Rewriting the sum as
$$\sum_{n=k+1}^{2k}\frac1n=\sum_{n=k+1}^{2k}\frac1k\cdot\frac1{n/k}$$
allows us to identify this as a Riemann sum related to the definite integral
$$\int_1^2\frac1x\,dx=\ln 2.$$
To see that, divide the interval $[1,2]$ to $k$ equal length subintervals, and evaluate the function $f(x)=1/x$ at the right end of each subinterval. When $k\to\infty$, the Riemann sums will then tend to the value of this definite integral.