, the Stolz-Cesàro theorem
is a criterion for proving the convergence
of a sequence
Let and be two sequences of real numbers. Assume that is positive, strictly increasing and unbounded and the following limit exists:
Then, the limit:
also exists and it is equal to
The Stolz-Cesàro theorem can be viewed as a generalization of the Cesàro mean, but also as a l'Hôpital's rule for sequences.
The theorem is named after mathematicians Otto Stolz and Ernesto Cesàro.