Undergraduate Analysis pp 206-245 | Cite as

# Series

Chapter

## Abstract

Let is called the call % MathType!MTEF!2!1!+-
its

*E*be a normed vector space. Let {*v*_{ n }} be a sequence in*E*. The expression$$\sum\limits_{n = 1}^\infty {{v_n}} $$

**series**associated with the sequence, or simply a series. We$${s_n} = \sum\limits_{k = 1}^n {{v_k}} = {v_1}\, + \,...\, + \,{v_n}$$

*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$$\sum\limits_{k = 1}^\infty {{v_k}} = \begin{array}{*{20}{c}} {\lim } \\ {n \to \infty } \end{array}{s_n}.$$

## Keywords

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