$\sum {a_n}$ be a convergent series of complex numbers but let $\sum |{a_n}|$ be divergent.. I am stuck on the following problem that says:

Let $\sum {a_n}$  be a convergent series of complex numbers but let $\sum |{a_n}|$
be divergent.
Then it follows that
a. $a_n \to 0$ but $|{a_n}|$  does not converge to $0$.
b. the sequence $\{a_n\}$ does not converge to $0$.
c. only finitely many $a_n$’s are $0$.
d. infinitely many $a_n$’s are positive and infinitely many are negative.
e. none of the above.

I am not sure which of the aforementioned options is correct . I think option (a) is the right choice but am unable to prove it. Wiki has been of little help.
Can someone explain? Thanks and regards.

1) Is not true. For example, take $a_n = \dfrac{(-1)^n}{n}$, $n > 0$.
2) Also not true. Take the same sequence.
3) Not true. Take $a_n = \dfrac{(-1)^n}{n}$ if $n$ is odd, and $0$ if $n$ is even. Then, $a_n$ converges and $|a_n|$ does not, but there are not finitely many $a_n$'s that are $0$.
4) True. If finitely many $a_n$'s are negative or positive, we can take them out and the series would be convergent, and since all of the would be positive or negative, the absolute value of the sequence would converge too.
For the fourth one, you could prove it more rigorously, but I let that for you.
