Stolz–Cesàro theorem

A limit theorem that, under suitable conditions, identifies the limit of a quotient of sequences with the limit of their successive differences.

Sequence: An ordered list of terms indexed by natural numbers or another ordered set. The theorem applies to pairs of sequences rather than functions.
Ratio test: A convergence test for series based on limits of ratios of consecutive term magnitudes. Stolz–Cesàro can justify ratio-limit calculations used in convergence arguments.
Finite difference: The change between values of a sequence at distinct indices. Successive differences play the role that derivatives play in continuous limit comparisons.
Otto Stolz: A German mathematician known for work in analysis and the foundations of calculus. Stolz's work gives one half of the theorem's standard name.
Limit of a sequence: A value that sequence terms approach as their indices grow. The theorem compares limits of difference quotients with limits of sequence quotients.
Cesàro mean: The average of the first terms of a sequence. Applying the theorem to partial sums and their index counts yields average-limit results.
Derivative: The instantaneous rate of change of a function with respect to its input. The theorem is a discrete analogue of comparing derivatives to evaluate quotient limits.
Ernesto Cesàro: An Italian mathematician whose work included summability and mathematical analysis. Cesàro's work gives the other half of the theorem's standard name.
Monotone sequence: A sequence whose terms never decrease or never increase. Monotonicity of the denominator sequence is a common sufficient hypothesis.
Harmonic series: The divergent series whose terms are the reciprocals of the positive integers. Its partial sums provide a standard example for estimating logarithmic growth with the theorem.

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