[Math] partial sum of series which is convergent but not absolutely convergent

sequences-and-series

I am stuck on the following exercise.

It is given that the series $\sum_{n=1}^\infty a_n$ is convergent, but not absolutely convergent and $\sum_{n=1}^\infty a_n=0$. Denote by $s_k$ the partial sum $\sum_{n=1}^k a_n$, k=1,2,… Then

  1. $s_k=0$ for infinitely many k

  2. $s_k>0$ for infinitely many k, and $s_k<0$ for infinitely many k

  3. it is possible that $s_k>0$ for all k

  4. it is possible that $s_k>0$ for all but a finite number of values of k

Here $\sum_{n=1}^\infty a_n=\lim_{k\to \infty}s_k=0$ hence it is possible that its partial sum $s_k=0$ for infinitely many k, hence first option is correct.

Here series is convergent but not absolutely convergent therefore this series has negative terms hence $s_k$ can be greater than and less than $0$ infinitely many times ,hence option 2 is correct.
I have example $\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}$ this series is convergent but not absolutely convergent and $s_k>0$ for all k for this series, hence option 3 is correct.
I am not getting option 4.

please correct me if i am wrong.

Thank you.

Best Answer

I am gonna assume you want sequences which satisfy the given conditions individually. Because some of them are contradictory to each other. So you can't find a single sequence satisfying all conditions. One more thing, the sequence you considered for option $3$, won't work out since it doesn't converge to $0$, which is one of the requirement.

Consider $ \{1,\,-1,\,1/2,\,-1/2,\, 1/3,\,-1/3...\}$ the sum converges to $0$, and doesn't converge absolutely, and the sum is $0$ for Infinitely $k$, for even numbered groups.

Say if you were to swap every other pair, then the sequence would be $ \{1,\,-1,\,1/2,\,-1/2,\, -1/3,\,1/3, \,-1/4, \,1/4...\}$, then for some infinite $k$, sum is negative for other infinite $k$ it is positive. So option $1,\, 2$ are possible.

Take $ \{-log\, 2, \, 1\, , -1/2\, , 1/3\, ,-1/4\}$, sum will be positive for all $k$, except for first one. The sum still goes to zero, not absolutely convergent. So option $4$ is possible. It should easy to find out why option 3 is not possible taking the same example, and modifying it little bit.

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