What is a formal example of a non-convergent Cauchy sequence in a metric space of rationals?

Question

With respect to a space $\mathbb{Q}$ of purely rational numbers with the standard metric $|p-q|$ and its properties, how does one prove a series like $$\sum_{n=0}^{\infty}\frac{(2k+1)!}{2^{3k+1}(k!)^{2}} \ \ \text{or} \ \ \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}$$ is Cauchy but does not converge in that space, without relying on knowing pre-hand that these converge to $\ln(2)$ and $\sqrt{2}$ respectively? This means if you were to address the problem, you would pretend only rational numbers exist and act as though you don't know any explicit values that are irrational.