Chapter 6 (Riemann/Stieltjes Integral)
Chapter 7 (Series of Real Numbers)
Homework
Math 555

section

note to yourself
of date due

problems
Riemann Integral
(RI)
handouts 		 
6.1
Riemann Integral
&
6.2
Properties of RI 		  A total of 4 problems.
  • 6.1.20
  • Pick 3 from:
    • 6.1: 7, 11, 13, 17, 18
    • 6.2: 4, 6, 7, 10
For these problems, you can use any of the 4 equivalent characterizations, that we showed in class, of a bounded RI function.
6.3
Fundamental Thm of Calculus 		  All of:
  • 7 (hint: no &epsilon's needed)
  • Read my blurb-in' and then do # 12
  • 13.   Read this important Warning !
  • The following strengthening of #16.
    Generalized Hölder's Inequality Let f,g ε ℜ [a,b]. Let 1 ≤ p1 , p2 , p3 < ∞ and (1/ p1 ) + (1/ p2 ) = (1/ p3 ) . Show that |fg|p3 , |f|p1 , |g|p2 ε ℜ [a,b] and
    || fg ||Lp3   ≤   || f ||Lp1   || g ||Lp2   .
    Remark/Hint: you may use (without proving) the usual Hölder's Inequality, which we showed in class (Theorem 2.5 from the Convex Function Handout).
6.4
Improper Riemann Integral 		  Read the section and the solution in back of book to Problem 6.4.9.a. Then do Problem 6.4.9 b/c/d.
6.5
Riemann-Stieltjes Integral 		  Read the section and then do Problem 6.5.3.
You can use (without proving) any theorem from this section.
6.6
Numerical Methods 		  We will skip.
6.7
Proof of Lebesgue's Thm 		  We already covered this section in the lectures on Sections 6.1&6.2. Hand in Problems: 6.7.2, 6.7.3, 6.7.4.
Chapter 6
Miscellaneous Exercises 		  1. (actually, problems 2 -- 5 are really niffty but I do not want to assign them because it would be too easy for you to look up solutions to them in any respectable differential equations book).
7.1
Convergence Tests
(positive termed series) 		  Read the section and then do Problem 20.
(Recall that you did Problems 15 and 16 way back when we covered 2.7.)
7.2
Dirichlet Test 		  Read the section and then do Problems: 1, 3.
7.3
Absolute and Conditional Convergence 		  Read the section and then do Problems: 1, 4.
7.4
Square Summable Sequences 		  Read the section and then do Problem: 12.
Note that we have basically covered most of the ideas in this section already. For Problem 12, you can use, without proving, anything from the class lectures on Convex Functions.
Chapter 7
Miscellaneous Exercises 		  5
Funky Functions
  Go to the website Interactive Real Analysis and link to Chapter 6 (Continuity and Differentiation) then to Section 6 (A Function Primer). There you will find Examples of Continuous and Differentiable Functions. Check out these 7 funky fuctions. Ok ... I need some way to know that you did this so tell me which one of the 7 funky functions is your favorite one.

Funky Functions Comments

8.1
Pointwise Convergence and Interchange of Limits 		  Read section and then do Problems 2, 3.

Findable from URL: http://www.math.sc.edu/~girardi/