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