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 \( \displaystyle \lim_{n\to\infty} \sqrt[n]{x_n} = \lim_{n\to\infty} \frac{x_n}{x_{n-1}} \).

• L'Hôpital's rule