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 $\sum_{n = 1}^{\infty} [\zeta(2n) - 1]$?

$$0\le \zeta(n)-1=\frac{1}{2^n}+\sum_{k\ge3}\frac{1}{k^n}\le\frac{1}{2^n}+\int_2^\infty\frac{dx}{x^n}=\frac{1}{2^n}+\frac{1}{(n-1)2^{n-1}}=\frac{n+1}{n-1}\frac{1}{2^n}$$
which $\le\frac{3}{2^n}$ for $n\ge2$. Thus, the sum converges by comparison with the geometric series $\sum \frac{1}{2^n}$.
