Homework, MATH 410, Fall 2012

Homework is from our text (Fitzpatrick) unless indicated.
Homework is due at the beginning of class on the day listed.

Homework #1 (Friday Sept. 7)

1.1: 1d, 2, 5, 11b [you may assume #10], 17
1.2: 4 [but you don't have to justify your answer]
1.3: 1,2,7,12,19 [you may use #15]

Homework #2 (Friday Sept. 14)

Continuity and uniform continuity with epsilon and delta .3: 25a (prove by induction)
2.1: 1,7,8,10,11,14,16,18

For #11,in each part give an example of a sequence satisfying the usual definition of convergence or the given definition, but not both definitions.)
Hint for #14: compute the first few sums to guess a formula, and prove by induction that for all n the formula holds for the sum of the first n terms)
Hint for #16: (a+b)(a-b)=a^2 - b^2. You can multiply expressions by suitable quantities x/x without changing the value of the expression ...
Above, e.g. a^2 means a to the second power, a times a.

2.2: 1,2
(For parts 1a,1b,1d,1e: if the statement is false, then you can just write a correct counterexample sequence, without justifying the counterexample.)
2.3: 8,9

Homework #3 (Friday Sept. 21)

2.3: 10,11
2.4: 1abcd and 2abce (write "True" without proof or write "False" and give a counterexample); 2d (prove this); 3a; 12 (for 12, to show boundedness of the sequence, you are allowed to be informal and simply draw the functional iteration picture for the appropriate function)
2.5: 1,7
Prove Proposition C on the handout "Open sets, closed sets and sequences of real numbers".
Prove "Associated facts" number 4 and number 6 on the "LIMINF and LIMSUP" handout.

Homework #4 (Friday Sept.28)

(Below: in the problem 1 in 3.1, 3.2 and 3.4: when the statement is false, you can write down a correct counterexample without justifying it.)
3.1: 1,4,14
3.2: 1,7
3.3: 6
3.4: 1, 5, 7
3.5: 1, 7c

Homework #5 (Wednesday Oct. 3)

3.6: 1, 8, 13
3.7: 2, 5, 11
4.1: 4a, 5ab, 9

Homework #6 (Friday Oct. 12)

4.2: 1, 2, 3
4.3: 5,19,20

Homework #7 (Friday Oct. 19)

6.1: 1, 3
6.2: 2 (In 2, just give an example showing the converse is not true.),3, 6a (You can assume the formula of 4a.), 10, 12, 13
6.3: 4 (just show the left inequality -- for sup -- of each case); also,
Problem X: is the upper integral of (f+g) always equal to (upper integral of f) + (upper integral of g)? Give a proof or counterexample.

Homework #8 (Friday Oct. 26)

6.4: 1,3,6, 7,9
6.5: 1, 6
6.6: 1bc,5,6,7 (just give an example for #7)
7.3: 1,2 (The game here is to recognize the sums as Riemann sums.)

Homework #9 (Friday Nov. 2)

8.1: 1bd,2ac
8.2: 1,3,10,11
8.3: 1b, 5 (Hint: show f satisfies (8.13).)
8.5: 8

Homework #10 (Friday Nov. 9)

8.6: 3
8.7: 4,5
9.1: 1 (for 1a, say for which pairs (a,p) the series converges), 3,7,8

Homework #11 (Wednesday Nov. 14)

9.2: 6
9.3: 1, 2,3
9.4: 2

Homework #12 (the last homework)(Monday Dec. 3)

9.4: 1
9.5: 1, 2, 6, 7, 8

Problem C*:

Let R denote the radius of convergence of a given power series a_0 + a_1x + a_2x^2 + ... .
Let L denote limsup |a_n|^{1/n} (where limsup is as n goes to infinity).
Here L could be zero, a positive real number, or +infinity.
Below, we define 1/L to mean zero if L is +infinity,
and we define 1/L to mean +infinity if L is zero.

Prove that R = 1/L .

*(Published by Cauchy in 1821. Yesterday's theorems are today's exercises ... )