ABSTRACT
Given any sequence of real numbers, one can construct from it a new sequence, called the sequence of Partial Sums. This notation works really well when the sequence starts at one. The only time the indexing of the partial sums matches the indexing of the original sequence is when the sequence starts at one. One can get a lot of information about a series by looking at the series one gets by summing the absolute values of the terms of the base sequence used to construct the partial sums. This chapter describes absolute convergence and conditional convergence. It provides basic facts about series, and the Cauchy criterion for series. The chapter describes the consequences of absolute convergence. A series that converges absolutely also converges. Series with non-negative terms, the completeness axiom, the Infimum tolerance Lemma, the Supreme tolerance Lemma are described. Bounded increasing or decreasing sequences converge, and non-negative series converge if and only if they are bounded.
