2
$\begingroup$

It is given that the series $\sum_{n=1}^\infty a_n$ is convergent, but not absolutely and $\sum_{n=1}^\infty a_n=0$. Denote by $S_k$ the partial sum $\sum_{n=1}^k a_n$ , $k=1,2,\dots$ Then,
(a) $S_k=0$ for infinitely many $k$;
(b) $S_k>0$ for infinitely many $k$ , $S_k<0$ for infinitely many $k$;
(c) it is possible that $S_k>0$ for all $k$;
(d) it is possible that $S_k>0$ for all but finite number of values of $k$.

I am completely stuck on it. How can I solve this problem? Please help.

  • 0
    Not sure I know where the answer below and its comment are aiming at, but anyway, (a) and (b) are false in general while (c) and (d) hold.2012-12-16

2 Answers 2

2

It helps to reformulate the assumptions in terms of $S_k$. We are told that

  • $S_k\to 0$
  • $\sum |S_{k+1}-S_k| =\infty$

and nothing else. Of course, there is nothing here that implies $S_k$ being zero, or positive for infinitely many values of $k$. The examples $S_k=(2+(-1)^k)/k$ and $S_k=(-2+(-1)^k)/k$ take care of all four parts, confirming Did's answer in a comment: "(a) and (b) are false in general while (c) and (d) hold".

1

Hint: Take the sequences $a_n=\begin{cases}\frac{1}{n}&\mbox{if, }n=2,4,6,...\\ \frac{-1}{n+1}&\mbox{if, }n=1,3,5,...\end{cases}$ and $b_n=\begin{cases}\frac{-1}{n}&\mbox{if, }n=2,4,6,...\\ \frac{1}{n+1}&\mbox{if, }n=1,3,5,...\end{cases}$ The series $\sum_{n=1}^{\infty}a_n=0=\sum_{n=1}^{\infty}b_n$ do not converge absolutely. This should help you rule out some options and point you in the right direction.

  • 0
    @gumti Yeah sorry. I was thinking about $a_n$ not $S_n$. (d) is easily verfied if you choose a sequence slightly different.2012-12-16