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

What is the sum of the ‘second half’ of the harmonic series?

$$\lim_{k\to\infty}\sum\limits_{n=k+1}^{2k}{\frac{1}{n}} =~ ?\lim_{k\to\infty}\sum\limits_{n=k+1}^{2k}{\frac{1}{n}} =~ ?$$

More precisely, what is the limit of the above sequence of partial sums?

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.