the number The partial sums themselves form a sequence , which is called the sequence of partial sums of the series . If the sequence of partial sums converges, then we say that the series is convergent, and the limit is called the value of the series. The same notation is used to denote a series and its value, i.e. we write . Use in other fields of mathematics Topology Sequences play an important role in topology, especially in the study of metric spaces. For instance: A metric space is compact exactly when it is sequentially compact. A function from a metric space to another metric space is continuous exactly when it takes convergent sequences to convergent sequences. A metric space is a connected space if and only if, whenever the space is partitioned into two sets, one of the two sets contains a sequence converging to a point in the other set. A topological space is separable exactly when there is a dense sequence of points. Sequences can be generalized to nets or filters. These generalizations allow one to extend some of the above theorems to spaces without metrics. Product topology The topological product of a sequence of topological spaces is the cartesian product of those spaces, equipped with a natural topology called the product topology. More formally, given a sequence of spaces , the product space is defined as the set of all sequences such that for each i, is an element of . The canonical projections are the maps pi : X → Xi defined by the equation . Then the product topology on X is defined to be the coarsest topology (i.e. the topology with the fewest open sets) for which all the projections pi are continuous. The product topology is sometimes called the Tychonoff topology. Analysis In analysis, when talking about sequences, one will generally consider sequences of the form which is to say, infinite sequences of elements indexed by natural numbers. A sequence may start with an index different from 1 or 0. For example, the sequence defined by xn = 1/log(n) would be defined only for n ≥ 2. When talking about such infinite sequences, it is usually sufficient (and does not change much for most considerations) to assume that the members of the sequence are defined at least for all indices large enough, that is, greater than some given N. The most