## Functional series |

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### Contents

Uniform Convergence | 1 |

Functional Properties of the Sum of a Series | 20 |

Applications | 56 |

Copyright | |

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### Common terms and phrases

Abel's test Abel's theorem absolute value absolutely convergent an(x analytic asymptotic expansion binomial series continuous convergent series converges uniformly defined derivatives differentiation Dirichlet's test divergent series end point Euler-Maclaurin formula Example expansion in powers fact Fichtenholz 2095 finite number fixed fn(x follows func function f(x functional series G. M. Fichtenholz given hence Hint immediate consequence infinite series integrated term interval of convergence limit function ln(l Moreover multiplying n=i n neighborhood nonuniform numerical series obtained P.P.Korovkin partial sums pn(x Pocket Mathematical Library point x0 polynomials power series expansion Prob Problem Proof Prove radius of convergence Rams Remark replacing right-hand side rn(x sequence of functions series 00 sufficiently small sum f(x Suppose the series take the limit Taylor series term by term test Theorem tion uk(x un(x uniform convergence vanishes variables Weierstrass whole interval write