I will now show that $\sum \frac{1}{2k}$ and $\frac{1}{2k+1}$ both diverge.
$\exists \ \epsilon > 0 \ \forall N \in \mathbb{N}$ so that for $m>n>N$:
$0\le |\sum_{k=n}^{m} \frac{1}{2k+1} | \le |\sum_{k=n}^{m} \frac{1}{2k}| < |\sum_{k=n}^{m} \frac{1}{k}| > \epsilon $ for some $n>N$
Tell me if this is formally correct. Please.