Suppose $\sum\limits_{n=1}^{\infty} a_n$ be a conditionally convergent series with $\sum\limits_{n=1}^{\infty} a_n = 0$ and the sequence of partial sums $\{S_k \}.$
Can it happen that $S_k > 0$ for all but a finite number of $k$?
I can't find any example where it holds. I tried by taking the series of sines and cosines and evaluate them at $\pi$ and $\frac {\pi} {2}$ respectively, where I saw that $S_k > 0$ for infinitely many $k$ and $S_k < 0$ for infinitely many $k.$ Is it true for all the conditionally convergent infinite series converging to $0$? If so, why? If not, can anybody present some counter-examples? Any help in this regard will be highly appreciated.
Thank you very much for your valuable time.
Best Answer
It is possible. Take $a_1=1$ and, for all $m\ge1$, take $a_{2m} = \frac1m$ and $a_{2m+1}=-\frac1m-\frac1{2^m}$. Then $$ S_k = \begin{cases} \displaystyle\frac1{2^{(k-1)/2}}, &\text{if $k$ is odd}, \\ \displaystyle\frac1{2^{k/2-1}}+\frac2k, &\text{if $k$ is even}, \end{cases} $$ which shows that all of the $S_k$ are positive and that $\sum_{n=1}^\infty a_n=\lim_{k\to\infty} S_k = 0$. On the other hand, $$ \sum_{n=1}^\infty |a_n| \ge \sum_{m=1}^\infty |a_{2m}| = +\infty, $$ and therefore $\sum_{n=1}^\infty a_n$ converges conditionally.