The

**Cauchy condition** for an

infinite series Σ a

_{n} on a

compact set is:

For all ε > 0, there exists an N such that for all m, n that satisfies m ≥ n > N,
| a_{n} +
a_{n+1} + … +
a_{m}| < ε

The series is said to be

**uniformly Cauchy** if it satisfies the Cauchy condition.

A series of complex numbers is

convergent if and only if it is uniformly Cauchy.