If formula_4 becomes arbitrarily large as formula_93, we write In this case we say that the sequence diverges, or that it converges to infinity. An example of such a sequence is . If formula_4 becomes arbitrarily negative (i.e. negative and large in magnitude) as formula_93, we write and say that the sequence diverges or converges to negative infinity. A series is, informally speaking, the sum of the terms of a sequence. That is, it is an expression of the form formula_98 or formula_99, where formula_54 is a sequence of real or complex numbers. The partial sums of a series are the expressions resulting from replacing the infinity symbol with a finite number, i.e. the "N"th partial sum of the series formula_98 is the number The partial sums themselves form a sequence formula_103, which is called the sequence of partial sums of the series formula_98. If the sequence of partial sums converges, then we say that the series formula_98 is convergent, and the limit formula_106 is called the value of the series. The same notation is used to denote a series and its value, i.e. we write formula_107. Sequences play an important role in topology, especially in the study of metric spaces. For instance:  Sequences can be generalized to nets or filters. These generalizations allow one to extend some of the above theorems to spaces without metrics. 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 formula_108, the product space  is defined as the set of all sequences formula_110 such that for each "i", formula_111 is an element of formula_112. The canonical projections are the maps "pi" : "X" → "Xi" defined by the equation formula_113. 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. 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. It may be convenient to have the sequence start with an index different from 1 or 0.