Proof for this sum?

Question

I was messing around with $$\sum_{n=2}^\infty (\zeta(n) -1)$$ and saw that it approaches 1. After putting it into wolfram, it says it does indeed converge to 1.

Like wise the sum $$\sum_{n=1}^\infty (\zeta(2n) -1) = \frac {3}{4} $$

I was wondering how these are proven? Or even if I am mistaken in assuming they're proven.

Answer 1

Note that $$\sum_{n=2}^\infty(\zeta(n)-1)=\sum_{n=2}^\infty\sum_{k=2}^\infty \frac1{k^n}.$$ Now reverse the order of summation....