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Erschienen in:
2009 | OriginalPaper | Buchkapitel
On convergent and divergent series. Rules for the convergence of series. The summation of several convergent series.
verfasst von : Prof. Robert E. Bradley, Prof. C. Edward Sandifer
Erschienen in: Cauchy’s Cours d’analyse
Verlag: Springer New York
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[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
.