The Cauchy condition
for an infinite series
on a compact
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.