Homework, MATH 410, Spring 2016

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

Homework #1 (Thursday Feb. 4)
(Note: due date changed due to the lost snow day)

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 (this is the "Reverse Triangle Inequality"),12,15

Homework #2 (Thursday Feb. 11)

1.3: 25a (prove by induction)
2.1: 1,7,8,11,14,16,18

For #1, "justify" with a proof or counterexample.
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 fracion s_n (in lowest terms) for the first few sums to guess a formula for s_n; then prove by induction that the formula holds for every s_n.
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; s_n means s with subscript n.

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 (Thursday Feb. 18)

2.3: 11 (For prob. 11, you may assume the result of problem 10.)
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. (Below: in the problem 1 in 3.1: when the statement is false, you can write down a correct counterexample without justifying it.)
3.1: 1,4,14

Homework #4 (Thursday Feb. 25)

(There are a couple of example problems worked here: Continuity and uniform continuity with epsilon and delta .)
(Below: in the problem 1 in 3.2, 3.3, 3.4 and 3.6: when the statement is false, you can write down a correct counterexample without justifying it.)
3.2: 1
3.3: 1,4
3.4: 1abc, 5, 9 (Hint: D = {1,2,3,...}), 11
3.5: 1(just at x=50), 4, 9
3.6: 1, 13

Homework #5 (Thursday March 3)

3.7: 2, 7
4.1: 1, 4a, 5d, 9
4.2: 1
4.3: 1,7,9,15 [hint: consider the function h(x)=f(x)/g(x)],19,20
4.4: 2

EXAM 1 will be Thursday March 10 on Chapters 1-4 and 5.1-5.2. There is no homework due this week. The following week (Monday March 14--Friday March 18) is spring break.

Homework #6 (Thursday March 24)

6.1: 1, 3
6.2: 2 (for 2, just give an example showing the converse is not true.),3, 6a (You can assume the formula of 4a.), 10, 12, 13
7.3: 1,2 (the game for 1 and 2 is to recognize the sums as Riemann sums), 10
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 #7 (Thursday March 31)

6.4: 1,6, 7,9
6.5: 1, 6
6.6: 1bc,5,6,7 (just give an example for #7)

Homework #8 (Thursday April 7)

8.1: 1bd,2ac,6
8.2: 1,3,5
8.3: 1b
8.5: 9
8.6: 1
8.7: 4,5

Homework #9 (Tuesday April 12)

9.1: 1bd, 2, 7,8

Homework #10 (Thursday April 21)

9.2: 6
9.3: 1,2,3
9.4: 1,2,4

Homework #11 (Thursday April 28)

9.5: 1, 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 ... )

(Hint.
CASE I: suppose 0 < |x|< 1/L. Then L < 1/|x|. Pick a positive real number s such that L < s < 1/|x|.
Then for all sufficiently large n, |a^{1/n}|< s, so |a_n||x|^n < s^n .
Compare with a tail of a geometric series.
CASE II: suppose 1/L < |x|. Then L > 1/|x| .
Therefore there are infinitely many n such that |a_n|^{1/n} > 1/|x|. For these n, |a_n||x|^n > 1 ... )