Week 6.

Monday, July 26. We proved the Mean Value Theorem, and proved that a function with f'>0 on (a,b) is monotonically increasing on (a,b). We proved Darboux's Theorem that derivatives enjoy the intermediate value property. (Here is a biography of Gaston Darboux. Many more mathematician's biographies can be found here.) We started discussing infinite series (chapter 4). We defined what it means for a series to converge to a (finite ) number, and proved that if a series converges, then given any epsilon>0 there is an N such that for all n>= N, |a_n| is less than epsilon. We looked at an example of a series that looked like it was converging but actually did not.

Tuesday, July 27. We discussed the Cauchy Criteria for convergence of an infinite series and showed that a series is convergent if and only if it satisifies the Cauchy Criteria. We defined absolute convergence and showed that a series which converges absolutely converges in the usual sense. We showed that an alternating series whose terms are decreasing in absolute value and going to zero converges.

Solutions to homework 4.

Homework due Tuesday, July 27: Section 3.4) 2, 3, 5, 10, 11, 14, 15, 22, 23, 26.

Wednesday, July 28. We proved the basic tests for convergence of infinite series: the comparison test, the root test, and the ratio test (chapter 4.1). We gave some examples of using these tests. We discussed some interesting questions that people had raised.

Thursday, July 29: We studied the convergence of power series. We introduced the idea of limit superior of a sequence of numbers and discussed some basic properties of it: if L = lim sup b_n, then 1) for all epsilon>0 we have b_n < L + epsilon for all but finitely many n. 2) for all epsilon>0 we have b_n>= L- epsilon for infinitely many n. We proved the existence of a radius of convergence R for power series using the concept of lim sup.

Friday, July 30. We discussed hypergeometric series and Gauss's test for convergence of hypergeometric series. The nice thing about this was that Gauss's test always returns a conclusive answer when dealing with hypergeometric series on the radius of convergence.

Homework due Friday, July 30: Section 3.5) 3, 4, 5, 7, 8, 9, 16.