Stolz–Cesàro theorem

Statement

Let $(a_n)$ and $(b_n)$ be real sequences and let $L$ be a real number. Assume that $(b_n)$ is strictly increasing and that $b_n \to +\infty$ as $n \to \infty$. Assume that $$\lim_{n \to \infty} \frac{a_{n+1} - a_n}{b_{n+1} - b_n} = L.$$ Then $a_n / b_n \to L$ as $n \to \infty$.

Remarks

There is also a $0/0$ form. If $a_n \to 0$, $b_n \to 0$, $(b_n)$ is strictly decreasing and the quotient of the differences tends to $L$, then $a_n / b_n \to L$. With $b_n = n$ it gives the convergence of the Cesàro means. For example, with $a_n = 1 + 2 + \cdots + n$ and $b_n = n^2$, the quotient of differences is $(n+1)/(2n+1) \to 1/2$, so $a_n / n^2 \to 1/2$. It is a discrete analogue of L'Hôpital's rule.

History

Otto Stolz proved the $\infty/\infty$ case in a book of 1885. Ernesto Cesàro gave it in an article of 1888.

Explanations

L1 human

To compare two sequences that grow without bound, it is enough to compare their steps. If each step of $a_n$ is about $L$ times the step of $b_n$, then $a_n$ is about $L$ times $b_n$ in the long run.