# Show that $S_n=0$ for infinitely many $n$ [duplicate]

Let the series $$\sum_{n=1}^{\infty}a_n$$ is convergent but not absolutely convergent and $$\sum_{n=0}^{\infty}a_n=0$$. Denote by $$S_k$$ the partial sums, Then prove that $$S_k=0$$ for infinitely many $$k$$.

My Efforts

Using the definition that a series converges if and only sequence of partial sum converges.

Now it is given that $$\sum_{n=0}^{\infty}a_n=0$$, it follows that sequence of partial sum converges to $$0$$.

Now we can only conclude that around each neighborhood of $$0$$ however small there exist a stage after which we have infinitely many $$S_k$$ belong to the neighborhood but that does not prove that $$S_k=0$$

I request for some helpful comments so that I can solve this problem.

• This seems a bit strong to be true, this would require the partial sums of the alternating harmonic series to be $ln(2)$ infinitely often right? Oct 1, 2018 at 2:22

I doubt if your assertion is correct. $$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...=\ln(2)$$ Therefore $$\ln(2)-1+\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-...=0$$. However $$\ln(2)$$ is irrational, while the partial sums of the rest of the series are all rational, so the entire partial sum can never $$=0$$.