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

  • Robert E. Bradley
  • C. Edward Sandifer
Chapter
Part of the Sources and Studies in the History of Mathematics and Physical Sciences book series (SHMP)

[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.

Keywords

Convergent Series Negative Term Positive Term Geometric Progression Integer Power 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceAdelphi UniversityGarden CityUSA
  2. 2.Department of MathematicsWestern Connecticut State UniversityDanburyUSA

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