What's the use of this theorem about series? Theorem:
If $\sum_{n=1}^{\infty} a_n$ converges, then $\lim_{n \to \infty} a_n = 0$. Therefore, if $\lim_{n \to \infty} a_n$ does not exist, or exists but is not zero, then the series $\sum_{n=1}^{\infty} a_n$ is divergent.
Now, I think the whole point of such a theorem is to use the limit to test if a series converges or diverges. But apparently, the converse of this theorem is false. We cannot conclude that if $\lim_{n \to \infty} a_n = 0$, then $\sum_{n=1}^{\infty} a_n$ converges. For example, the harmonic series is a counter example to this. 
But I guess my question then is: why is there such a theorem if it doesn't go in both ways? I mean, if I know that my series converges, then I automatically know that the limit of the series will be zero. That's not really interesting. I would want the limit to give me information about my series, but that's not possible here, because if the $n$th term approaches $0$ as $n$ approaches $\infty$, then the series may or may not converge.
 A: There are multiple convergence tests that can check if a series converges. The "holy grail" would be a test that is easy, works on every sequence, and is never inconclusive (it always says if the series converges or does not converge).
Unfortunately, there is no such test. The main tests, the ratio, root, and the currently discussed test (limit of the summand), all have inconclusive cases. In our current test, if the limit of the summand is zero, the test is inconclusive. If the ratio of consecutive terms or $n$'th root of the $n$'th term tend to one those tests are inconclusive.
Some tests, like the integral test, have no inconclusive case. However, they cannot reasonably be applied to every series. Too many integrals are not known! To use the comparison tests you need another series that is already known and relevant. And so on.
There is no perfect convergence test. My first-year analysis class had a devious series of homework problems designed to press that idea into our brains. Look at the good side: there is job security for mathematicians!
To sum up: This test works often enough to be useful. That is why it is used, even if it does not always help. It is one more tool in your toolbox.
