Category Archives: Series of Functions

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

Posted in Absolute Convergence, Analysis, Conditional Convergence, M-Test, Math, Series, Series of Functions, Uniform Convergence | Leave a comment

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

Posted in Analysis, Compact, Math, Riemann Integrable, Series of Functions, Uniform Convergence | Leave a comment

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

Posted in Analysis, Cauchy, Math, Sequence, Series of Functions, Uniform Convergence | 1 Comment