Category Archives: Uniform Convergence

UIC Master’s Exam- Fall 2007 R1

Problem Statement: Consider the power series (a) For which values of does it converge absolutely? Conditionally? (b) Show that on the convergence is uniform. Solutions: (a) Claim: the series converges absolutely on and conditionally on . Consider , then it follows that … Continue reading

Winter 2008 #5

Problem Statement: Let . Prove is Riemann Integrable on and .  Proof: First note that each is continuous on and since is a sum of continuous functions it follows that is continuous on . Furthermore, since is a compact set it … Continue reading

Spring 2008 #5

Problem Statement: Let be a sequence of continuous functions defined on and a real valued function defined on . (a) Prove if uniformly then and (b) Does this hold is pointwise? Prove or give a counter example. (a) Proof: We are given … Continue reading

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Winter 1993

Problem Statement: Suppose that converges to uniformly and converges to uniformly on where each is continuous on . Prove that converges uniformly to on . Proof: Since is a compact set and are each continuous on it follows that each is bounded … Continue reading

Winter 1995

Problem Statement: Let and let for all and for every . Show that if converges then converges uniformly on . Note: This is asking us to prove the Weierstrass M-Test. Proof: Let and for every and for every . Assume that converges. … Continue reading

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Winter 1995 #4

Problem Statement: Let be uniformly continuous and let for every . Show converges uniformly. Proof: First I claim that converges pointwise to . Let . Then . Since is continuous and converges to it follows that . This holds for all … Continue reading

Spring 1996 #4

More studying of sequences and series of functions. These boogers are beginning to be a little more manageable. Problem Statement: Consider the sequence of functions for . (a) Prove converges. (b) Is the convergence uniform? Prove your answer. (a) I claim that converges pointwise to … Continue reading

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