SOLUTIONS TO THE SAMPLE MIDTERM 1
1. Prove that
1
1
1
n
1
+
+
+ +
=
.
12 23 34
n(n + 1)
n+1
Solution. Let P (n) be the statement
1
1
1
1
n
+
+
+ +
=
.
12 23 34
n(n + 1)
n+1
The statement P (1) is true
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:
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): Ap
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
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
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).
W
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
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
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
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 characterizatio
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.
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
Pr
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
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 =
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 achiev
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, w
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
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) li
Homework #2
3.1 (a) Which of the properties A1A4, M1M4, DL, O1O5 fail for N?
(b) Which of these properties fail for Z?
Solution: (a) Since 0 N, A3 and A4 have to fail. (That A4 fails is also clear sin
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
Undergraduate Texts in Mathematics
StephenAbbott
Understanding
Analysis
Second Edition
Undergraduate Texts in Mathematics
Undergraduate Texts in Mathematics
Series Editors:
Sheldon Axler
San Francisco
SOLUTIONS TO HOMEWORK 5
Warning: Essentially no proofreading has been done.
1. Section 3.2
Problem 1.1 (Problem 3.2.1). (a) In the proof of Theorem 3.2.3, part (ii) we
choose
= mincfw_ 1 , , N ,
the m
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
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
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
SOLUTIONS TO HOMEWORK 7
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 3.2
Problem 1.
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.
SOLUTIONS TO HOMEWORK 2
N. CHRISTOPHER PHILLIPS
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
SOLUTIONS TO HOMEWORK 5
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
Pr
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