MTH 436 Spring 06 -- Tips for Exam 2

MTH 436 Spring 06 -- Tips for Exam 2
Exam 2 is scheduled for Friday, April 21, 3-5, room TBA (soon).

Exam 2 covers classes 35-44 and homework assignments 5-7. You are expected to know and be able to state all definitions, propositions, and theorems given in class and in homework assignments as well as their proofs. You should know all of the important examples studied in class. You are expected to know solutions to all homework problems. Some problems on the exam will ask you to state and prove a theorem proved in class; some will ask you to state a definiton and give an example; some will be similar to homework problems.

While I was browsing my notes and homework assignments, the following questions jumped at me as good candidates for exam questions. (I am indicating below theorems, propositions, examples studied in class without stating precisely all the assumptions. Consult you notes for that.)

-- State the 2nd FTC. Prove something like Corollary 35.1, Corollary 35.2.
-- State the definiton of pointwise convergence of a sequence of functions f_n to a function f on I.
-- State the definiton of uniform convergence of a sequence of functions f_n to a function f on I.
-- Give an example of a sequence of functions that converges pointwise but not uniformly.
-- Give an example of a sequence of continuous functions converging pointwise to a discontinuous limit. Under what assumptions is continuity preserved?
-- State the theorem about convergence of integrals of a uniformly convergent sequence of funtions.(Th 37.2 or Th. 38.1) Prove the easier case (Th 37.2).
-- Give an example of a sequence of functions converging pointwise such that the integrals of terms do not converge to the integral of the limit.
-- State the Cauchy condition for uniform convergence (Th 38.2). Prove the theorem.
-- State and prove the necessary condition for convergence of a series of constants (Th. 40.3).
-- Show that the harmonic series diverges.
-- State and prove the Limit Comparison Test (Th 41.3).
-- Show that an absolutely convergent series is convergent (Th 43.2).
-- Define pointwise and uniform convergence for a series of functions.
-- Establish a region of covergence for a given series of functions (something like Ex 44.1).
-- Give an example of a series which converges pointwise but not uniformly.
-- State and prove the theorem about integrating a series term-by-term (Th 44.2).

Of homework problems, the following seem particularly good:

H4: #2, #4, #5;
H5: All are excellent candidates;
H6: #1, #3, #7, #8;
H7: #1, #5, #6.

Consider it a practice exam (Of course, the actual exam will be much shorter). Some of the problems listed above will appear, some will not. A problem or two not listed above may appear as well. Who knows. I hope you all get As!