Activity 8.3.5.
Consider the harmonic series \(\sum_{k=1}^{\infty} \frac{1}{k}\text{.}\) Recall that the harmonic series will converge provided that its sequence of partial sums converges. The \(n\)th partial sum \(S_n\) of the series \(\sum_{k=1}^{\infty} \frac{1}{k}\) is
\begin{align*}
S_n =\mathstrut \amp \sum_{k=1}^{n} \frac{1}{k}\\
=\mathstrut \amp 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n}\\
=\mathstrut \amp 1(1) + (1)\left(\frac{1}{2}\right) + (1)\left(\frac{1}{3}\right) + \cdots + (1)\left(\frac{1}{n}\right)\text{.}
\end{align*}
Through this last expression for \(S_n\text{,}\) we can visualize this partial sum as a sum of areas of rectangles with heights \(\frac{1}{m}\) and bases of length 1, as shown in the image below, which uses the 9th partial sum.
The graph of the continuous function \(f\) defined by \(f(x) = \frac{1}{x}\) is overlaid on this plot.
(a)
Explain how this picture represents a particular Riemann sum.
(b)
What is the definite integral that corresponds to the Riemann sum you considered in (a)?
(c)
Which is larger, the definite integral in (b), or the corresponding partial sum \(S_9\) of the series? Why?
(d)
If instead of considering the 9th partial sum, we consider the \(n\)th partial sum, and we let \(n\) go to infinity, we can then compare the series \(\sum_{k=1}^{\infty} \frac{1}{k}\) to the improper integral \(\int_{1}^{\infty} \frac{1}{x} \ dx\text{.}\) Which of these quantities is larger? Why?
(e)
Does the improper integral \(\int_{1}^{\infty} \frac{1}{x} \ dx\) converge or diverge? What does that result, together with your work in part 8.3.5.d, tell us about the series \(\sum_{k=1}^{\infty} \frac{1}{k}\text{?}\)
