Evaluate $\sum \sum 1/n^k $

I wanted to evaluate the sum:

$$ \sum_{n \ge 2} \left(\zeta(n) – 1\right) $$

I rewrote this as:

$$ \sum_{n\ge 2} \sum_{k\ge 2} \frac{1}{n^k} $$

I tried exploiting the symmetry but that didn’t seem to help. I know from numerical calculation that the answer is 1.

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