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.

  • 1
    $\begingroup$ 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? $\endgroup$
    – TomGrubb
    Oct 1, 2018 at 2:22

1 Answer 1


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 $=0$.


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