The nth partial sum Sn is the sum of the first n terms of the sequence; that is,
A series is convergent (or converges) if the sequence of its partial sums tends to a limit; that means that, when adding one after the other in the order given by the indices, one gets partial sums that become closer and closer to a given number. More precisely, a series converges, if there exists a number such that for every arbitrarily small positive number , there is a (sufficiently large) integer such that for all ,
If the series is convergent, the (necessarily unique) number is called the sum of the series.
The same notation
is used for the series, and, if it is convergent, to its sum. This convention is similar to that which is used for addition: a + b denotes the operation of adding a and b as well as the result of this addition, which is called the sum of a and b.
Any series that is not convergent is said to be divergent or to diverge.