Show that $$\sum_{k=2}^\infty (\zeta(k) - 1)$$ converges and find the limit.
(Hint : Deduce that $\zeta(s) = 1 + O(2^{-\sigma})$ where $s = \sigma + it$, $\sigma \geq 2$.)
$\textbf{Proof}$ Let $s = \sigma + it$ with $\sigma \geq 2.$ Then $$\zeta(s) = 1 + \sum_{n \geq 2} \frac{1}{n^s} = 1 + \sum_{n \geq 2} \frac{1}{n^\sigma e^{it \log n}}.$$
Since $s$ is complex and complex number cannot be compared, is it a bit vague to prove $$\zeta(s) = 1 + O(2^{-\sigma}).$$ Even if I ignore the imaginary part, it is not clear why $$\sum_{n \geq 2} \frac{1}{n^\sigma}\leq \frac{C}{2^\sigma}$$ for some constant $C$.