• Serge Lang
Part of the Undergraduate Texts in Mathematics book series (UTM)


Let E be a normed vector space. Let {v n } be a sequence in E. The expression
$$\sum\limits_{n = 1}^\infty {{v_n}} $$
is called the series associated with the sequence, or simply a series. We
$${s_n} = \sum\limits_{k = 1}^n {{v_k}} = {v_1}\, + \,...\, + \,{v_n}$$
call % MathType!MTEF!2!1!+- its n-th partial sum. If \(\begin{array}{*{20}{c}} {\lim } \\ {n \to \infty } \end{array}\) s n exists, we say that the series converges, and we define the infinite sum to be this limit, that is
$$\sum\limits_{k = 1}^\infty {{v_k}} = \begin{array}{*{20}{c}} {\lim } \\ {n \to \infty } \end{array}{s_n}.$$


Power Series Uniform Convergence Open Interval Pointwise Convergence Convergent Series 
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 New York 1997

Authors and Affiliations

  • Serge Lang
    • 1
  1. 1.Department of MathematicsYale UniversityNew HavenUSA

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