<s>[INST] 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?  
 [/INST] 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. 
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