Category Archives: Series

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

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 | Leave a comment

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

Posted in Analysis, Cauchy, Math, Sequence, Series | Leave a comment

May 1999 #1

Problem Statement: Let be a sequence of real numbers. Prove that converges if and only if converges. Proof:  Assume that converges to some real number . Let . Then there exists an such that for every . Consider . . Let , … Continue reading

Posted in Analysis, Sequence, Series | Leave a comment

Cauchy Condensation Test

The Cauchy Condensation Test states that for a decreasing sequence where for all then converges if and only if converges. Proof: In the forward direction, let us assume that converges. This means that the sequence of partial sums, converge. Define … Continue reading

Posted in Analysis, Math, Sequence, Series | Leave a comment

September 1999 #1

Problem Statement: Let be a sequence of non-negative real numbers. Assume there exists a real number such that converges. Show converges. Proof: Let be a sequence of non-negative real numbers. We know there exists a real number such that converges. Since the … Continue reading

Posted in Analysis, Cauchy, Comparison Test, Math, Sequence, Series | Leave a comment

September 2001 #2

Problem Statement: Suppose that converges absolutely. Prove that converges for every . Proof: We are given that  converges absolutely, this implies that  converges. By definition this means that the sequence of partial sums, , converges. But every convergent sequence is also Cauchy. Let … Continue reading

Posted in Analysis, Cauchy, Sequence, Series | Tagged , , | 3 Comments