Assume that the limit of the sequence is zero, $\lim_{n\to\infty}a_n=0$. So its not plainly obvious if the series $\sum a_n$ converges or diverges.
I have wondered for some time. If $\lim_{n\to\infty}a_n=0$ then it must be the case that $\lim_{n\to\infty}a_{n+1}=0$ and $\lim_{n\to\infty}a_{n+2}=0$ and so on.
Does it not make sense then that $\lim_{n\to\infty}a_n + a_{n+1} + a_{n+2}=0$
Cauchy's Convergence test basically says this, does it not? $$\lim_{\substack{m\to\infty\\n\to\infty} }\sum_{k=m}^{m+n}a_k = 0$$
I realize that Cauchy's Convergence Criterion is a necessary and sufficient condition for convergence and divergence, while the Divergence test is just necessary for convergence. That said, I am wondering why. Why doesnt meeting the divergence test imply meeting Cauchy's convergence criterion?