Course Play by Play
Wk 1: Wed Jan 23: Intro to course. Problems in the world. Real Analysis can help solve (some of) them. Dynamical systems worksheet. Iteration. What topic from Math 301 does this remind you off?
For Friday, finish worksheet up through example (4): f(x) = ex.
Fri Jan 25: Review of worksheet. What happens to iteration of different initial values? Make conjectures about possible behaviors. Counter-example to conjecture. List of analysis terms related to dynamical systems:
- sequence, bounded or unbounded sequence, (monotone) increasing or decreasing sequence, convergence, limit, continous function, subsequences, Bolzano-Weierstrass Theorem (any bounded sequence has a converge subsequence), Monotone Convergence Theorem (a bounded, monotone sequence converges – useful for proving existence of limits! ), divergence (either to infinity or via oscillation). Cauchy sequences.
We did not cover all the topics that are on the hw sheet for next week so only do the following:
#1 using absolute value notation (not with distance notation); #4.
Wk 2: Monday Jan 28. Dr. Francl guest lecture on Quantum Mechanics.
Wed Jan 30: Review definition of limit of sequence in R using absolute value notation. Rewrite this definition using distance notation. Consider sequences in R2, limit of sequence, definition using notion of standard Euclidean distance in R2 (Rn).
Metric space (B: Sect 11.4). Please finish homework worksheet 1 for Friday.
Friday Feb 1: Definition of limit in a metric space. Visualization of Epsilon nbd in R2. Example of function space C[0,1]. Explorations of what might make a good distance between functions (metric); worksheet. Formal definition of metric space.
Wk 3: Mon Feb 4. Using the definition of metric, proved that R2 with the Euclidean distance is a metric. Proved theorem relating limit in R2 to limits in R. Taxi cab metric, discrete metric.
Web Feb 6: Review of Cauchy sequences. Complete, non-complete metric spaces. Open, closed balls in metric spaces. Quiz.
Wk 4: Mon Feb 11: Home work extension from Wed until Friday. Discussion of how to use triangle inequality from R to prove that triangle inquality holds for d_infinity and d_1 metrics. Limits of sequences of functions. Group worksheet.
Wed Feb 13: Whole space, null set are both open and closed. Discussion of pointwise convergence of sequence of functions vs uniform convergence.
Friday Feb 15: Uniform convergence. How to prove non-uniform convergence.
Wk 5: Mon Feb 18: Review of continuity. Delta-epsilon cheer. Continuity of functions between metric spaces. Uniform limit of continuous functions is a continuous function.
Wed Feb 20: Equivalence of uniform convergence and convergence in the sup metric. Taylor polynomials as an example of sequence of functions. Taylor polynomial Mathematica notebook.
Friday Feb 22: More Taylor polynomials for ex and the regions on which they are good approximations to the ex.
Wk 6: Mon Feb 25: Taylor Remainder Theorem and application to prove pointwise and uniform convergence.
Wed Feb 27: Review for test. See exam review outline guide.
Friday Feb 29: Defintion review with groups. Review of pointwise, uniform convergence with epsilon proofs.
Wk 7: Mon March 4: Introduction to infinite series. Limit of partial sums. Geometric series.
Wed March 5: Use Mathematica to evaluate partial sums of series. Conjecture whether the limit will exist or not (ie whether series converges). Mathematica commands for series, sums.
Friday March 7: test due. Cauchy Criterion, Comparison Test.
Wk 8: Mon March 17: Alternating Harmonic Series as example of alternating series.
Wed March 19: Absolutely convergent series are convergent.
Friday March 21: review of key ideas about series and typical confusions. Favorite examples of series that illustrate various properties. Start mini-presentation process (see instructions).
Wk 9: Mon March 24
Wed March 26: Proof of convergence of Sum (1/n^2) using comparison with geometric series. Student presentations on various series tests. Root test.
Friday March 28: Ratio and Limit Comparison test.
Wk 10: Mon March 31: Integral test. Feedback questionnaire.
Wed April 2: Intuition behind conv/div of series
Frid April 4: Series of Functions. (Sect 9.4)
Wk 11: Mon April 7: M- test,
Wed April 9: Theorem of uniform convergence of sequence of functions, proofs
Fri April 11: radius of convergence of power series, interchanging integral and limit.
Wk 12: Mon April 14: Loose ends: limits of form n1/n ; interchange limit and integral; radius of convergence of power series, uniform convergence.
Wed April 16: review of topics for midterm.