Does $\sum_{n = 2}^{\infty} [\zeta(n) - 1]$ converge? Question in the title. Does $\sum_{n = 2}^{\infty} [\zeta(n) - 1]$ converge? If not, how about $\sum_{n = 1}^{\infty} [\zeta(2n) - 1]$?

Figured out the solution:$\sum_{n=2}^{\infty} [\zeta(n) - 1] = \sum_{n=2}^{\infty}\sum_{k=2}^{\infty} n^{-k} = \sum_{n=2}^{\infty}\frac{1}{1-\frac{1}{n}} - 1 - \frac{1}{n} = \sum_{n=2}^{\infty}\frac{1}{n(n-1)}$, which converges by comparison with $\zeta(2)$.
