Homework #6
14.3 Determine
(a)
(c)
(e)
which of the following series converge. Justify your answers.
1/ n!
(b)
(2 + cos n)/3n
1/(2n + n)
(d)
(1/2)n (50 + 2/n)
sin(n/9)
(f)
(100)n /n!
Solution: (a): Applying the Ratio Test, we obtain
lim sup
1/ (n + 1)!
an
University of Oregon: Math 315
Final Exam: June 12, 2009 Friday
Show all your work or you may not get credit. Use lots of space.
1. (15 Points) Let (sn ) be a sequence. State the following denitions:
n=1
(a) (sn ) converges to s.
(b) (sn ) is Cauchy.
(c)
SOLUTIONS TO HOMEWORK 2
In some cases, the statement of the problem has been rewritten to be more
precise.
Warning: Essentially no proofreading has been done.
1. Section 1.4
Problem 1.1 (Problem 1.4.1).
Solution. This just show in the proof of Theorem 1.4
SOLUTIONS TO HOMEWORK 3
In some cases, the statement of the problem has been rewritten to be more
precise.
Warning: Essentially no proofreading has been done.
1. Section 2.3
Problem 1.1 (Problem 2.3.1). Show that the constant sequence (a, a, a, a, . . .)
SOLUTIONS TO HOMEWORK 1
In some cases, the statement of the problem has been rewritten to be more
precise. Also, occasionally a missing hypothesis has been added (usually that
some set is nonempty).
Warning: Essentially no proofreading has been done.
1. S
Homework #8
19.1 Which of the following continuous functions are uniformly continuous on the
specied set? Justify your answers.
(a) f (x) = x17 sin(x) ex cos(3x) on [0, ].
(b) f (x) = x3 on [0, 1].
(c) f (x) = x3 on (0, 1).
(d) f (x) = x3 on R.
(e) f (x)
Math 315
Pelatt Summer 2012
Homework One Solutions
1.2.1 (a) Prove
that 3 is irrational. Does a similar argument work to
show 6 is irrational?
Proof. First, we claim that if n N and n2 is divisible by 3, then
n is also divisible by 3. We prove the contrap
Math 315
Platt Summer 2012
Homework Seven Solutions
2.7.1 Proving the Alternating Series Test amounts to showing that the sequence of partial sums
sn = a1 a2 + a3 an
converges. Dierent characterizations of completeness lead to dierent
proofs.
(a) Prove th
Math 315
Pelatt Summer 2011
Homework Eight Solutions
3.2.1 (a) Where in the proof of Theorem 3.2.3 part (ii) does the assumption
that collection of open sets be nite get use?
Solution. We need N to be nite so that we can say that =
min(cfw_1 , 2 , . . . ,
SOLUTIONS TO HOMEWORK 3
N. CHRISTOPHER PHILLIPS
In some cases, the statement of the problem has been rewritten to be more
precise.
Warning: Essentially no proofreading has been done.
1. Section 1.3
Problem 1.1 (Problem 1.3.4). Assume that A and B are none
SOLUTIONS TO HOMEWORK 4
N. CHRISTOPHER PHILLIPS
In some cases, the statement of the problem has been rewritten to be more
precise.
Warning: Little proofreading has been done.
1. Section 1.3
Problem 1.1 (Problem 1.3.5b). For A R and c R, dene
cA = cfw_ca :
SOLUTIONS TO HOMEWORK 7
Warning: Essentially no proofreading has been done.
1. Section 4.4
Problem 1.1. Problem 4.4.4.
Proof. Since f is continuous on [a, b], hence by extremum value theorem, f achieves
its maximum and minimum on [a, b]. Let m = minx[a,b]
Math 315
Pelatt Summer 2012
Homework Five Solutions
2.3.9 Does Theorem 2.3.4 remain true if all of the inequalities are assumed
to be strict? If we assume, for instance, that a convergent sequence
(xn ) satises xn > 0 for all n N, what may we conclude abo
Math 315
Pelatt Summer 2011
Homework Two Solutions
1.3.2 (a) Write a formal denition in the style of Denition 1.3.2 for inmum or greatest lower bound of a set.
Solution. Let A R and let s R. Then s = inf(A) if, rst, s
is a lower bound for A, and, second,
Homework #9
23.1 For each of the following power series, nd the radius of convergence and
determine the exact interval of convergence.
(a)
(c)
(e)
(g)
n2 xn
(2n /n2 )xn
(2n /n!)xn
(3n /n 4n )xn
(b)
(d)
(f)
(h)
(x/n)n
(n3 /3n )xn
(1/(n + 1)2 2n )xn
(1)n /n
SOLUTIONS TO HOMEWORK 6
Warning: Essentially no proofreading has been done.
1. Section 4.2
Problem 1.1 (Problem 4.2.1ab). Use Denition 4.2.1 to supply proofs for the
following limit statements.
(1) limx2 (2x + 4) = 8.
(2) limx0 x3 = 0.
Solution. (1) Let >
Undergraduate Texts in Mathematics
StephenAbbott
Understanding
Analysis
Second Edition
Undergraduate Texts in Mathematics
Undergraduate Texts in Mathematics
Series Editors:
Sheldon Axler
San Francisco State University, San Francisco, CA, USA
Kenneth Ribet
Math 315
Pelatt Summer 2011
Homework Six Solutions
2.5.1 Prove Theorem 2.5.2:
Theorem 0.1 (2.5.2). Subsequences of a convergent sequence converge
to the same limit as the original sequence.
Proof. Let (an ) be a convergent sequence, and call the limit a.
Math 315
Pelatt Summer 2012
Homework Three Solutions
2.2.1 Verify, using the denition of convergence of a sequence, that the following sequences converge to the proposed limit.
(a)
lim
Proof. First, we note that
1
= 0.
+1
6n2
1
6n2 +1
< 1 for all n N. Thu
MATH 315 SPRING 2015: MIDTERM 2 VERSION 2 SOLUTIONS
N. CHRISTOPHER PHILLIPS
1. (5 points.) State the Nested Interval Property (Theorem 1.4.1 of the book).
Solution. For each n Z>0 , let In = [an , bn ] R be a nonempty closed interval.
Assume that
I1 I2 I3
SOLUTIONS TO HOMEWORK 9
N. CHRISTOPHER PHILLIPS
In some cases, the statement of the problem has been rewritten to be more
precise.
Warning: Very little proofreading has been done.
1. Section 4.3
Problem 1.1 (Problem 4.3.8).
(1) Show that if a real valued
SOLUTIONS TO HOMEWORK 10
N. CHRISTOPHER PHILLIPS
In some cases, the statement of the problem has been rewritten to be more
precise.
Warning: Little proofreading has been done.
1. Section 5.3
Problem 1.1 (Problem 5.3.8). Let a, b R with a < b and let c (a,
MATH 315 WINTER 2015: MIDTERM 1 SOLUTIONS
N. CHRISTOPHER PHILLIPS
Warning: Little proofreading has been done.
1. (5 points.) State the Nested Interval Property (Theorem 1.4.1 of the book).
Solution. For each n Z>0 , let In = [an , bn ] R be a nonempty clo
MATH 315 (PHILLIPS) QUIZ 1 (Friday 16 January 2015).
NAME: Solutions
Student id: -
INSTRUCTIONS: Show all work, and use correct notation. Closed book; this includes: No
notes and no electronic devices of any kind (calculators, cell phones, dictionaries, i
MATH 315 (PHILLIPS) QUIZ 3 (Friday 13 February 2015).
NAME: Solutions
Student id: 2 3 2 4 2- 3 3 3- 5 3 5 4 5 5 5
INSTRUCTIONS: Show all work, and use correct notation. Total 30 points. Time: 25
minutes.
1. (4 points.) State the denition of a Cauchy seq
MATH 315 (PHILLIPS, WINTER 2015) HOMEWORK 1
SOLUTIONS
Warning: Essentially no proofreading has been done.
1. Convergence of series
Problem 1.1. For each of the following series, use the methods of Math 253 to
show, as appropriate, that it converges or tha
MATH 315 WINTER 2015: FINAL EXAM SOLUTIONS
N. CHRISTOPHER PHILLIPS
Warning: Almost no proofreading has been done.
1. (5 points.) State the Axiom of Completeness.
Solution. Let A R be a subset which is nonempty and bounded above. Then A
has a least upper b
SOLUTIONS TO HOMEWORK 6
N. CHRISTOPHER PHILLIPS
In some cases, the statement of the problem has been rewritten to be more
precise.
Warning: Essentially no proofreading has been done.
1. Section 2.3
Problem 1.1 (Problem 2.3.10). Let (an )nZ>0 and (bn )nZ>0
MATH 315 (PHILLIPS) QUIZ 2 (Friday 23 January 2015).
NAME: Solutions
Student id: -
INSTRUCTIONS: Show all work, and use correct notation. Closed book; this includes: No
notes and no electronic devices of any kind (calculators, cell phones, dictionaries, i
SOLUTIONS TO HOMEWORK 8
N. CHRISTOPHER PHILLIPS
In some cases, the statement of the problem has been rewritten to be more
precise.
Warning: Little proofreading has been done.
1. Section 3.3
Problem 1.1 (Problem 3.3.2). Prove the converse of the part of Th