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An absolutely convergent series has Cauchy partial sums.
Absolute convergence implies Cauchy: Let be a series of real or complex numbers, and let be its partial sums. If
converges, then is a Cauchy sequence; equivalently, for every there exists such that for all integers ,
This is the series form of the Cauchy criterion and is a standard route to absolute convergence implies convergence.
A partial sum is a finite sum associated to a series .
The sequence encodes the series: statements about convergence or divergence of the series are statements about whether the partial sums form a sequence with a limit, as formalized in convergent series.
Cauchy criterion in : Let be a sequence in , with distance measured by the Euclidean norm (equivalently, the Euclidean metric). Then converges in if and only if it is a Cauchy sequence, meaning: for every there exists such that for all , .
Absolute convergence implies convergence: If a series is absolutely convergent, meaning that
converges, then the original series converges.