In this chapter we investigate whether we can “add” infinitely many real numbers. In other words, if
$$\sum\nolimits_{n = 1}^\infty {a_n }$$
is a sequence of real numbers, then we ask whether we can give a meaning to a symbol such as “
$$a_1 + a_2 + \cdots + a_n + \cdots$$
” or
$$\left( {a_n } \right)_{n \ge 1}$$
. We are mainly concerned with series of positive numbers, alternating series, but also with arbitrary series of real numbers, viewed as a natural generalization of the concept of finite sums of real numbers. This leads us to discuss deeper techniques of testing the behavior of infinite series as regards convergence. The theory of series may be viewed as a sequel to the theory of sequences developed in Chapter 1.


Positive Integer Real Number Prime Number Positive Real Number Qualitative Result 
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Copyright information

© Springer-Verlag New York 2009

Authors and Affiliations

  1. 1.Department of MathematicsFratii Buzesti National CollegeCraiovaRomania
  2. 2.Simion Stoilow Mathematics InstituteRomanian AcademyBucharest Netherlands
  3. 3.School of Natural Sciences and MathematicsUniversity of Texas at DallasRichardsonUSA

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