# partial sum of convergent series

Let series $\sum^\infty a_n$ is convergent but not absolutely convergent. And $\sum^\infty a_n =0$. Denote $s_k$ the partial sum $\sum_{n=1}^k a_n$, $k=1,2,\dots$ then which of following ARE true.

1.$s_k=0$ for infinitely many $k$.

1. $s_k \gt 0$ for infinitely many $k$, and $s_k \lt 0$ for infinitely many $k$

2. It is possible that $s_k \gt 0$ for all $k$

3. It is possible that $s_k \gt 0$ for all but finite number of values of $k$.

I think series could be alternating one, converging to zero. So its partial sum sequence converges to zero. The partial sum sequence won't be monotonic, because it's not a positive term series, that's why I think option 2 is true and option 3 should be false. but about other options I don't know. Whats should be the correct options?

• 3. is possible. Start with $a_1=1$. Take the next $9$ terms to be $-.1$. Take the $11$'th term to be $.4$, to give a partial sum of $1/2$. Then take small negative terms till you get close to $0$. Then take the next term approximately $1/3$, to give a partial sum of $1/3$. And so on. – David Mitra Jan 29 '15 at 12:52
• Whst about 1 and 4, I think atleast they cant be true. Isn'Isn't it – singularity Jan 29 '15 at 13:03
• In my last comment, rather than "taking a bunch of small steps back to $0$", you could just take one big step. 4. can hold since 3. can. 1. and 2. need not hold since 3. can hold. – David Mitra Jan 29 '15 at 13:47
• Is there a reason why 2 cannot hold? – singularity Jan 29 '15 at 13:51
• A series satisfying 3. would not satisfy 2. So 2. is false in general. There are series, though, that satisfy 2. – David Mitra Jan 29 '15 at 15:15

$$s_k=\sum_{n=1}^k\frac {(-1)^{n+1}} n$$
So first $s_k$ is convergent (this is a well known result, you can try to group the terms two-by-two), but not absolutely convergent (harmonic series, also a classic result).
1. $\forall k\in\mathbb N,s_k>0$ (intuitively, at each step, either you increase $s_k$ or you decrease by less than what you added at the previous step), so 1. is false in general