.
Well your example only mean the smallest term is of the order of $\frac{1}{n^2}$ and for the largest term, the nth term, one has the indeterminate $0\cdot \infty$. A brief check on their derivatives wrt n both exists, hence we can use L hopital and found that it becomes $-\frac{1}{(n-k)^2}$ thus tends to zero as $n\to \infty$. Therefore your series should converge since any partial sum is between $\frac{1}{n^2}$ and $0$, which by p series test they both converge thus by sandwich theorem, the sum shoudl converge to zero

Btw guys, a possible group theory question. Suppose I have a set of 3 elements $S=\{1,2,3\}$ and the permutations defined by multiplication as $1=1_{S_3}$, $2=(23)$ and $3=(32)$. Their action on $S$ seemed to form a structure where $(23)$ and $(32)$ forms a group wrt the subset $\{2,3\}\subset S$ which is isomorphic to $\mathbb{Z/2}$ and $1_{S_3}$ act like the identity as usual.
But it is clear that the overall $(S,\cdot)$ is not a group, yet it look like a direct product of two groups. Is this direct product $(S,\cdot)\cong \{1\}\times \mathbb{Z/2}$?