If I take an $= (-1)^{\frac{(n+1)}n}$ then series of these terms is convergent by Leibnitz and not absolutely but I'm not able to get sum is zero. Please help me how to solve this problem. Please share a link if it is already solved on this site.
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 of $\sum _{n=1} ^ k a_n ,k-1,2,\dotsc $. Then
(a.) $S_k = 0$ for infinitely many values of $k$.
(b.) $S_k \gt 0$ for infinitely many values of $k$ and $S_k \lt 0$ for infinitely many values of $k$..
(c.) It is possible for $S_k \gt 0$ for all $k$.
(d.) It is possible for $S_k \gt 0$ for all but a finite number of $k$.