Understanding when $\lim_\limits{n \to \infty} \frac{a_n}{b_n} = \lim_\limits{n \to \infty} \frac{a_0 + a_1+ \cdots + a_n}{b_0+b_1+ \cdots + b_n}$.
How to understand intuitively the Stolz-Cesaro Theorem for sequences?