Cauchy's convergence test

Cauchy's convergence test states that a series converges if and only if its partial sums form a Cauchy sequence. It characterizes convergence through the smallness of every sufficiently late finite tail.

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Cauchy criterion: A metric-space criterion stating that a sequence converges exactly when it is Cauchy in a complete space. The series test applies this general criterion to the sequence of partial sums.
Infinite series: A sequence of terms considered through the limit of its finite partial sums. The test decides convergence of this object by examining its partial sums.
Alternating harmonic series: The series whose terms alternate in sign and have magnitudes equal to reciprocals of the positive integers. Its tails can be bounded directly, establishing convergence without absolute convergence.
Term test for divergence: The necessary condition that terms of a convergent series tend to zero. Unlike the Cauchy criterion, it can disprove convergence but cannot establish it.
Cauchy sequence: A sequence whose terms become arbitrarily close to one another as their indices grow. Partial sums must satisfy this condition for the series to converge.
Sequence convergence: Convergence of a sequence to a limit in a specified space. A series converges precisely when its sequence of partial sums converges.
Uniform convergence: Convergence of functions in which one index threshold works uniformly across the domain. A uniform Cauchy criterion tests it by bounding differences of function-sequence terms.
Ratio test: A series test based on the limit or limit superior of ratios of successive term magnitudes. It is often more convenient, but may be inconclusive where the Cauchy criterion still applies.
Partial sum: The sum of the first finitely many terms of a series. The test checks whether these sums become mutually close in their tails.
Completeness: The property that every Cauchy sequence in a space has a limit within that space. Completeness explains why the Cauchy condition on partial sums ensures a sum exists.