Answer
Calculate the value of the harmonic sums
$1+ \frac{1}{2}+ \frac{1}{3} + \cdots + \frac{1}{n}$ for $n=2,3,4,5$.
For $n=2$,
$1+\frac{1}{2} = 3/2 = 1.5$
For $n=3$,
$1+ \frac{1}{2}+ \frac{1}{3} = 11/6 = 1.8333...$
For $n=4$,
$1+ \frac{1}{2}+ \frac{1}{3} + \frac{1}{4}= 25/12 = 2.08333...$
For $n=5$,
$1+ \frac{1}{2}+ \frac{1}{3} + \frac{1}{4}+ \frac{1}{5}= 137/60= 2.28333...$
Work Step by Step
Recall from calculus that the area underneath the graph of $y=1/x$ between $x=1$ and $x=n$ is $\ln (n)$, where $\ln (n) = \log_e(n)$. Thus the harmonic sum approximates $\ln(n)$.
