What you easily get for any integer k >= 2$k \ge 2$ is the sum of 1/n^k$\sum 1/n^k$ summed over all nonzero integers n$n$, both positive and negative. When k$k$ is even, we can use this to evaluate zeta(k)$\zeta(k)$. When k$k$ is odd, we get the (correct, but uninteresting) result 0$0$.