0.301 The series

1.

$\sum _{k=1}^{\infty}{f}_{k}}\left(x\right),$

the terms of which are functions, is called a functional series. The set of values of the independent variable x for which the series 0.301 1 converges constitutes what is called the region of convergence of that series.

0.302 A series that converges for all values of x in a region M is said to converge uniformly in that region if, for every e = 0, there exists a number N such that, for n > N, the inequality

$\left|{\displaystyle \sum _{k=n+1}^{\infty}{f}_{k}}\left(x\right)\right|<\epsilon $

holds for all x in M.

0.303 ...

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