Divergent series whose terms converge to zero I've just begun to teach my class series.  We usually have workshops where they work in groups on a tougher problem, and I was thinking of asking them to come up with a divergent series whose terms converge to $0$. While $\sum 1/n$ is the prototypical example, we haven't done the integral test yet.  Do you know of any simpler examples?  

The harmonic series is pretty simple and you do not need the integral test to show it diverges. Here is a short proof by contradiction:
Suppose $\sum^{\infty } _{n=1}\frac{1}{n}$ converges to a finite number $L$. Then, 
$L=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\cdots +\cdots$ and this is greater than
$1+\frac{1}{2}+\frac{1}{4}+\frac{1}{4}+\frac{1}{6}+\frac{1}{6}+\frac{1}{8}+\frac{1}{8}+\cdots$. Now group the terms:
$1+\frac{1}{2}+\left ( \frac{1}{4}+\frac{1}{4} \right )+\left ( \frac{1}{6}+\frac{1}{6} \right )+\left ( \frac{1}{8}+\frac{1}{8} \right )+\cdots $ and this latter item is
$1+\frac{1}{2}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots =L+\frac{1}{2}$ so what we've shown is 
$L\geq L+\frac{1}{2}$ which is absurd. Hence, the series does not converge. 
