A student studies the sums Sn = 1/(n + 1) + 1/(n + 2) + … + 1/2n, which have n terms…

A student studies the sums $S_n = \dfrac{1}{n + 1} + \dfrac{1}{n + 2} + \dots + \dfrac{1}{2n}$, which have $n$ terms. She remembers that the harmonic series $1 + \dfrac{1}{2} + \dfrac{1}{3} + \dots$ grows without limit, so she is not sure that sums of this kind stay bounded. She wants to know whether $S_n$ settles on a number as $n$ grows. She first finds $S_{10}$ and $S_{100}$ with a calculator, before she tries any other idea. Then she writes each term as $\dfrac{1}{n + r} = \dfrac{1}{n} \times \dfrac{1}{1 + r/n}$, so that $S_n = \dfrac{1}{n}\displaystyle\sum_{r=1}^{n}\dfrac{1}{1 + r/n}$. This is a right-hand sum for a definite integral on $[0, 1]$. She replaces $\dfrac{r}{n}$ by $x$ and $\dfrac{1}{n}$ by $dx$, and then evaluates the integral with a logarithm. Her teacher asks her to explain every step in plain words. Find $S_{10}$ and $S_{100}$ to four decimal places, and the limit of $S_n$ as $n \to \infty$ to four decimal places.