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

  • Robert E. Bradley
  • C. Edward Sandifer
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.


Convergent Series Negative Term Positive Term Geometric Progression Integer Power 
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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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