How to find the values of the entries of partial sums of a conditionally convergent series?

Question

It is given that the series $\sum_{n=1}^\infty a_n$ is convergent, but not absolutely convergent and $\sum_{n=1}^\infty a_n=0$. Denote by $s_k$ the partial sum $\sum_{n=1}^k a_n,k=1,2,.....$. Then

1. $s_k$=0 for infinitely many $k$.
2. $s_k>0$ for infinitely many $k$, and $s_k<0$ for infinitely many $k$.
3. It is possible that $s_k>0$ for all $k$.
4. It is possible that $s_k>0$ for all but a finite number of values of $k$. I know that such a series is conditionally convergent and I tried to find a related theorem or any kind of statement but I couldn't so I do not have the idea for approaching this problem. Please help. (p.s.. I know that this is a duplicate question but those answers could not help me, the correct answer is most probably (2))

Answer 1

Hint: For an example where all $s_n > 0$, try $s_n = 1/n$ for $n$ even and $2/n$ for $n$ odd.