On convergent and divergent series. Rules for the convergence of series. The summation of several convergent series.

[114]We call a series an indefinite sequence of quantities, u0, u1, u2, u3, …, which follow from one to another according to a determined law. These quantities themselves are the various terms of the series under consideration. Let be the sum of the first n terms, where n denotes any integer number. If, for ever increasing values of n, the sum s _n indefinitely approaches a certain limit s, the series is said to be convergent, and the limit in question is called the sum of the series. On the contrary, if the sum s _n does not approach any fixed limit as n increases indefinitely, the series is divergent, and does not have a sum. In either case, the term which corresponds to the index n, that is u _n , is what we call the general term. For the series to be completely determined, it is enough that we give this general term as a function of the index n.