partial sum of series which is convergent but not absolutely convergent I am stuck on the following exercise.  
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  


*

*$s_k=0$ for infinitely many k  

*$s_k>0$ for  infinitely many k, and $s_k<0$ for infinitely many k  

*it is possible that $s_k>0$ for all k  

*it is possible that $s_k>0$ for all but a finite number of values of k  
Here $\sum_{n=1}^\infty a_n=\lim_{k\to \infty}s_k=0$ hence it is possible that its partial sum $s_k=0$ for infinitely many k, hence first option is correct.
Here series is convergent but not absolutely convergent therefore this series has negative terms  hence $s_k$ can be greater than and less than $0$ infinitely many times ,hence option 2 is correct.
I have example $\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}$ this series is convergent but not absolutely convergent and $s_k>0$ for all k for this series, hence option 3 is correct.
I am not getting option 4.
please correct me if i am wrong.
Thank you.
 A: I am gonna assume you want sequences which satisfy the given conditions individually. Because some of them are contradictory to each other. So you can't find a single sequence satisfying all conditions. One more thing, the sequence you considered for option $3$, won't work out since it doesn't converge to $0$, which is one of the requirement.
Consider $ \{1,\,-1,\,1/2,\,-1/2,\, 1/3,\,-1/3...\}$ the sum converges to $0$, and doesn't converge absolutely, and the sum is $0$ for Infinitely $k$, for even numbered groups.
Say if you were to swap every other pair, then the sequence would be    $ \{1,\,-1,\,1/2,\,-1/2,\, -1/3,\,1/3, \,-1/4, \,1/4...\}$, then for some infinite $k$, sum is negative for other infinite $k$ it is positive. So option $1,\, 2$ are possible.
Take $ \{-log\, 2, \, 1\, , -1/2\, , 1/3\, ,-1/4\}$, sum will be positive for all $k$, except for first one. The sum still goes to zero, not absolutely convergent. So option $4$ is possible. It should easy to find out why option 3 is not possible taking the same example, and modifying it little bit.
A: *

*Take $a_n=\dfrac{(-1)^n}{n}$.


Then $s_1=-1,s_2=\frac{-1}{2}$,...
We have $s_n<0$ for all $n$. So A is false .


*B is false also


3.Take $a_n=\dfrac{(-1)^{n+1}}{n}$.
Then $s_n>0$ for all $n$.
