Conditions for generalized convergence I am a little bit curious about the following question:
Is it true that if an infinite series $\sum_{n\in\mathbb{N}}a_n$ neither converges to a real number nor diverges to $\infty$ (i.e., it is vibrating), then $\lim_{n\to\infty}a_n\ne 0$?
Can anyone give me some hints to show this or a counterexample to kill this proposition? Many thanks!
 A: It is not true. Here is a counterexample: 
Recall that $\sum_n\frac1n=\infty$. We have that $\frac1n\to0$, and that $\sum_{n>k}\frac1n=\infty$ for all $k$. 
Now, pick two numbers $s<L$ (for "small" and "Large"), and consider a sequence $a_1,a_2,a_3,\dots,$ where $a_i=1/i$ or $a_i=-1/i$ for all $i$. We choose the sign according to the following rules: $a_1,a_2,\dots,a_k$ are positive until
 $$\sum_{n\le k}a_n=\sum_{n\le k}\frac1n>L.$$ 
Then $a_{k+1},a_{k+2},\dots,a_t$ are all negative, until 
 $$ \sum_{n\le t}a_n=\sum_{n\le k}\frac1n-\sum_{n=k+1}^t\frac1n<s. $$
Then $a_{t+1},a_{t+2},\dots,a_r$ are all positive, until 
 $$ \sum_{n\le r}a_n=\sum_{n\le k}\frac1n-\sum_{n=k+1}^t\frac1n+\sum_{n=t+1}^r\frac1n>L. $$
Etc. That we can keep doing this forever follows from the fact that all tails of the harmonic series diverge, as mentioned above.
The series so defined diverges, but the sequence of partial sums has a subsequence that converges to $L$, and one that converges to $s$. (In fact, $L$ is the limit superior of the partial sums, and $s$ is the limit inferior.) In particular, the series does not diverge to $\infty$ (nor does it oscillate unboundedly). And since $|a_n|=1/n$ for all $n$, the $a_n$ indeed converge to $0$. 
