Real Analysis – Convergent but Not Absolutely Convergent Series $\sum a_n$ with $\sum_{n=1}^{\infty} a_n=0$

Question

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?

Answer 1

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.