Stolz–Cesàro theorem

Stolz–Cesàro theorem is a powerful tool for evaluating limits of sequences, and it is a discrete version of L'Hôpital's rule.

Theorem 1 (the $\infty/\infty$ case):
Let $(a_n)_{n\in\mathbb N}$ and $(b_n)_{n\in\mathbb N}$ be two sequences of real numbers such that
(a) $0<b_1 < b_2 < \cdots < b_n < \cdots$ and $\displaystyle \lim_{n\to\infty} b_n = \infty$ .
(b) $\displaystyle \lim_{n\to\infty} \dfrac{a_{n+1} - a_n}{b_{n+1} - b_n} = l \in \mathbb R$ .

Then, $\displaystyle \lim_{n\to\infty} \dfrac{a_n}{b_n}$ exists and is equal to $l$ .

Theorem 2 (the $0/0$ case):
Let $(a_n)_{n\in\mathbb N}$ and $(b_n)_{n\in\mathbb N}$ be two sequences of real numbers such that
(a) $\displaystyle \lim_{n\to\infty} a_n = \lim_{n\to\infty} b_n = 0$ .
(b) $(b_n)$ is strictly decreasing.
(c) $\displaystyle \lim_{n\to\infty} \dfrac{a_{n+1} - a_n}{b_{n+1} - b_n} = l \in \mathbb R$ .

Then, $\displaystyle \lim_{n\to\infty} \dfrac{a_n}{b_n}$ exists and is equal to $l$ .

Theorem 3 (Reciprocal of Stolz–Cesàro lemma):
Let $(a_n)_{n\in\mathbb N}$ and $(b_n)_{n\in\mathbb N}$ be two sequences of real numbers such that
(a) $0<b_1 < b_2 < \cdots < b_n < \cdots$ and $\displaystyle \lim_{n\to\infty} b_n = \infty$ .
(b) $\displaystyle \lim_{n\to\infty} \dfrac{a_n}{b_n} = l \in \mathbb R$ .
(c) $\displaystyle \lim_{n\to\infty} \dfrac{b_n}{b_{n+1}} = L \in \mathbb R\backslash \{1\}$ .

Then, $\displaystyle \lim_{n\to\infty} \dfrac{a_{n+1} - a_n}{b_{n+1} - b_n}$ exists and is equal to $l$ .

Theorem 3 (Multiplicative property):
If the sequence $(x_n /x_{n-1} )_{n=1}^\infty$ has a limit, then $\displaystyle \lim_{n\to\infty} \sqrt[n]{x_n} = \lim_{n\to\infty} \frac{x_n}{x_{n-1}}$ .