Let $S=\sum_{n=1}^\infty a_n$ be a convergent (but not absolutely convergent) series with $S=0$. Let $S_k=\sum_{n=1}^ka_n$ be the partial sum ($k\geq 1$).
What are examples of:
- $S_k=0$ for infinitely many $k$
- $S_k>0$ for infinitely many $k$ and $S_k<0$ for infinitely many $k$
- $S_k>0$ for all $k$
- $S_k>0$ for all but a finite number of $k$
I am not able to think of an example for the given situations. Please help me, I am a beginner.
All sequences start at index $1$ as given in the question.
For (i), use $a_{2p} = \frac{-1}{p}$ and $a_{2p-1} = \frac{1}{p}$.
For (ii), with $b_n= \frac{(-1)^{n+1}}{\sqrt{n+1}}-\frac{(-1)^n}{\sqrt{n}}$, it is clear that the series alternates (using telescoping series ).
Take $(c_k)_{k\geq 0}$ so that $c_n=\frac{1}{\sqrt{n+1}}-\frac{1}{\sqrt{n}}$, again using telescoping series, you have an example for (iii) and (iv) (for (iv), just offset the sequence with zeroes at the begininng).