m3115 - Uniform Convergence and Differentiability Series of...

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Uniform Convergence and Differentiability Series of Functions Power Series Taylor Series Workshop #10 Math 311: Advanced Calculus Wolmer V. Vasconcelos Set 5 Spring 2007 Wolmer Vasconcelos Set 5 Advanced Calculus
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Uniform Convergence and Differentiability Series of Functions Power Series Taylor Series Workshop #10 Main Goal Understand Sequences and Series of Functions Wolmer Vasconcelos Set 5 Advanced Calculus
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Uniform Convergence and Differentiability Series of Functions Power Series Taylor Series Workshop #10 Outline 1 Uniform Convergence and Differentiability 2 Series of Functions 3 Power Series 4 Taylor Series 5 Workshop #10 Wolmer Vasconcelos Set 5 Advanced Calculus
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Uniform Convergence and Differentiability Series of Functions Power Series Taylor Series Workshop #10 Sequences of Functions Let f n : A R , n N , be a set of functions. For each x A they define a numerical sequence ( f n ( x )) . If f n ( x ) L , we say that ( f n ) converges at x . We are greatly interested in case it converges to all x A , as the limit f n ( x ) f ( x ) will define a function f : A R . 1 If the f n are continuous, when is f continuous? 2 If the f n are differentiable, when is f differentiable? Wolmer Vasconcelos Set 5 Advanced Calculus
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Uniform Convergence and Differentiability Series of Functions Power Series Taylor Series Workshop #10 Example Let f n ( x ) = x n , n N , be the sequence of powers of x as functions on [ 0 , 1 ] . For any x in this interval, we have lim n →∞ f n ( x ) = 0 , 0 x < 1 lim n →∞ f n ( x ) = 1 , x = 1 Thus lim n →∞ f n exists for all x [ 0 , 1 ] , but it is not a continuous function on the interval. We need a rule that guarantees that lim n →∞ f n is continuous. Wolmer Vasconcelos Set 5 Advanced Calculus
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Uniform Convergence and Differentiability Series of Functions Power Series Taylor Series Workshop #10 Pointwise and Uniform Convergence Definition The sequence of functions ( f n ( x )) converges pointwise to f ( x ) if for every x f n ( x ) converges to f ( x ) . For a given x , this means that given ± > 0 there is N = N ( x ) N such that for n N , | f n ( x ) - f ( x ) | < ± . Another definition of convergence is much more restrictive: Definition The sequence of functions ( f n ( x )) converges uniformly to f ( x ) if for every ± > 0 there exists N N such that for n N , | f n ( x ) - f ( x ) | < ±. Wolmer Vasconcelos Set 5 Advanced Calculus
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Uniform Convergence and Differentiability Series of Functions Power Series Taylor Series Workshop #10 Example: Let f n ( x ) = 1 / n ( 1 + x 2 ) . Then f ( x ) = lim n →∞ f n ( x ) = 0. Given ± > 0 | f n ( x ) - f ( x ) | < 1 / n Thus if N 1 , | f n ( x ) - f ( x ) | < ± for n N . Wolmer Vasconcelos Set 5 Advanced Calculus
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Uniform Convergence and Differentiability Series of Functions Power Series Taylor Series Workshop #10 Cauchy Criterion for Uniform Convergence Theorem A sequence of functions ( f n ( x )) defined on a set A R converges uniformly on A if and only if for every ± > 0 there exists N N such that | f n ( x ) - f m ( x ) | < ± for all n , m N and all x A.
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m3115 - Uniform Convergence and Differentiability Series of...

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