Sequences and Series, Domination

Domination

The sequence s dominates the sequence t if s and t are positive and sj ≥ tj for each index j. The same definition holds if s and t are series.

If the series s dominates the series t, and s converges, then each partial sum in t lies below the sum of s. These partial sums form an increasing, bounded sequence, and by the previous theorem, t converges.