TITLE: $\sum a_n$ be convergent but not absolutely convergent, $\sum_{n=1}^{\infty} a_n=0$
QUESTION [11 upvotes]: Let $\sum a_n$ be convergent but not absolutely convergent, $\sum_{n=1}^{\infty} a_n=0$,$s_k$ denotes the partial sum then could anyone tell me which of the following is/are correct?
$1.$ $s_k=0$ for infinitely many $k$
$2$. $s_k>0$ and $<0$ for infnitely many $k$
3.$s_k>0$ for all $k$
4.$s_k>0$ for all but finitely many $k$
if we take $a_n=(-1)^n{1\over n}$ then $\sum a_n$ is convergent but not absolutely convergent,but I don't know $\sum_{n=1}^{\infty} a_n=0$? so I am puzzled could any one tell me how to proceed?

REPLY [7 votes]: None of them are necessarily true.
We can easily compute a series from its partial sums, so let's specify the $s_k$.
Define
$$
s_k=\left\{\begin{array}{}
-\frac1k&\text{if $k$ is odd}\\[4pt]
-\frac1{k^2}&\text{if $k$ is even}
\end{array}\right.
$$
Then $a_1=-1$ and for $k\gt1$,
$$
a_k=\left\{\begin{array}{}
\frac1{(k-1)^2}-\frac1k&\text{if $k$ is odd}\\[4pt]
\frac1{k-1}-\frac1{k^2}&\text{if $k$ is even}
\end{array}\right.
$$
Show that this series is not absolutely convergent, its sum is $0$, and it fails to satisfy any of the conditions.