### Blogroll

### Archives

- August 2011 (16)
- June 2011 (5)
- May 2011 (30)
- April 2011 (4)

### Categories

Absolute Convergence Algebra Analysis Cauchy Center Compact Comparison Test Continuity Countable Cyclic Dense Differentiable Field Group Integral Domain Linear Operator Math MVT Order Principal Ideal Riemann Integrable Sequence Sequence of Functions Series Series of Functions Squeeze Theorem Topology Uncategorized Uniform Continuity Uniform Convergence

# 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

## 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

## 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