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# Category Archives: Analysis

## UIC Master’s Exam- Fall 2007 R2

Problem Statement: is continuous on . Given that (a) prove has continuous derivative on (Hint: start with the change of variable u=x+t). (b) Given and show that there exists a such that for every . Solutions: (a)Proof: First let us start … Continue reading

## 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

## Rudin: Ch. 5 #4

Problem Statement: If where are real constants, prove that has at least one real root between and . Proof: Consider the polynomial on . Then it follows that and which equals by our assumption. Furthermore, since is a polynomial with real coefficients … Continue reading

Posted in Analysis, Differentiable, Math, MVT
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## Suz and Mike had a question

Problem Statement: If is bounded with finitely many discontinuities on then is Riemann Integrable on . Proof: Let be the number of discontinuities of on and let be such that for every . We know such an exists since is bounded. Since … Continue reading

Posted in Analysis, Math, Riemann Integrable
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## Rudin- Ch. 3 #13

Problem Statement: Given and define the product to be where . Suppose that converges to absolutely and converges to absolutely. Prove that converges to a value absolutely. Proof: Since and converge absolutely we know that their product converges to (this is by … Continue reading

Posted in Absolute Convergence, Analysis, Math, Sequence, Series
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## Rudin: Ch. 3 #8

Problem Statement: If converges and is monotonic and bounded, prove that converges. Proof: Since we are given that is bounded and monotonic we know there exists some such that . Let , then there exists some such that for every . Note 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