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,
| an +
an+1 + … +
am| < 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.