m3115 - Uniform Convergence and Differentiability Series of...

This preview shows pages 1–9. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 04/02/2008 for the course MATH 311 taught by Professor Vasconceles during the Spring '08 term at Rutgers.

Page1 / 54

m3115 - Uniform Convergence and Differentiability Series of...

This preview shows document pages 1 - 9. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online