About Riemann-Zeta function (2). Inspired by and dedicated to Sambhrant Sachan

Two proofs:

$\boxed{1.-}$ $\large \displaystyle \zeta (k) > 1, \space (\forall k \ge 2, \space k \in \mathbb{N}) \Rightarrow \sum_{k =2}^\infty \zeta(k) = + \infty$

$\boxed{2.-}$

If $\displaystyle \sum_{k = 2}^\infty \zeta(k)$ converged then $\displaystyle \lim_{k\to \infty} \zeta(k) = 0$ , but $\displaystyle \lim_{k\to \infty} \zeta(k) = 1$ , so the previous sum doesn't converge and therefore, due to $\boxed{1}$ it diverges to $+\infty$