According to the $n$th-term test for divergence, a series $\sum_{n=1}^\infty a_n$ diverges if $\lim_{n\to\infty} a_n \neq 0$.
But I don't actually get it.
I thought that if I have $$ \lim_{N\to\infty} \sum_{n=1}^N a_n = C $$ then the series $\sum_{n=1}^n a_n$ converges against $C$? Isn't this correct?
So I thought that if $$ \lim_{n\to\infty} a_n = C $$ where $C$ also can be $0$, the value of the series is $C$? Is this correct?
I have maybe misunderstood something about convergence of series.
You have to make sure you aren't confusing the $a_n$'s, which you are summing up, with the sequence of partial sums, $a_1$, $a_1+a_2$, $a_1+a_2+a_3$, etc.
Indeed, if the series is convergent, then the sequence of partial sums converges to some limit. But if $a_n$ doesn't go to $0$ for large $n$, then you are summing up an infinite number terms who are either constant and non-zero, or even increasing. Thus the infinite sum will "blow up."