 Such a series 
formula_30 is considered to be convergent if and only if the sequence of partial sums formula_31 is convergent, where 
formula_32.It is a routine matter 
to determine whether the sequence of partial sums is Cauchy or not,
since for positive integers "p" > "q", 

If formula_34 is a uniformly continuous map between the metric spaces "M" and "N" and ("x""n") is a Cauchy sequence in "M", then formula_35 is a Cauchy sequence in "N".If formula_36 and formula_37 are two Cauchy sequences in the rational, real or complex numbers, then the sum formula_38 and the product formula_39 are also Cauchy sequences.