Mathematics 3790H Analysis I: Introduction to analysis
Trent University, Fall 2009
Solution to Assignment #1
In solving the following problem, you may assume without further ado that for any
x > 0 and
Mathematics 3790H Analysis I: Introduction to analysis
Trent University, Fall 2010
Solutions to Assignment #2
Ces`ro Salad?
a
Ernesto Ces`ro (18591906) was an Italian mathematician who worked in the e
Mathematics 3790H Analysis I: Introduction to analysis
Trent University, Fall 2008
Assignment #3
The integral form of the remainder of a Taylor series
Suppose that a is a real number and f (x) is a fu
Mathematics 3790H Analysis I: Introduction to analysis
Trent University, Fall 2010
Solutions to Assignment #1
Eulers Constant
Eulers constant is the real number dened by:
n
= lim
n
k=1
1
k
ln(n) = l
Mathematics 3790H Analysis I: Introduction to analysis
Trent University, Fall 2009
Solutions to Assignment #2
For questions 1 and 2, assume that we know that
x2
x3
xn
x
+
+ =
e =1+ +
1!
2!
3!
n!
n=0
x
Mathematics 3790H Analysis I: Introduction to analysis
Trent University, Winter 2012
Solutions to Assignment #6
More p-tests
1. Determine for which p the series
ln(n)
converges and for which it diverg
Mathematics 3790H Analysis I: Introduction to analysis
Trent University, Winter 2012
Solutions to Assignment #7
Find the limit!
1
1
and an+1 =
for n 0.
2
1 + an
Show that this sequence converges and n
Mathematics 3790H Analysis I: Introduction to analysis
Trent University, Winter 2014
Take-home Final Exam
Due on Friday, 18 April, 2014.
Instructions: Do all three of parts , , and , and, if you wish,
Mathematics 3790H Analysis I: Introduction to analysis
Trent University, Winter 2014
Solutions to the Quizzes
Quiz #1. Tuesday, 14 January, 2014. [10 minutes]
1
1. Suppose you are given that inf n | n
Mathematics 3790H Analysis I: Introduction to analysis
Trent University, Winter 2012
Solution to Assignment #5
Squeezing more out of the Integral Test
1
diverged by interpreting the series as a sum of
Mathematics 3790H Analysis I: Introduction to analysis
Trent University, Winter 2012
Solution to Assignment #2
an
L
=
. [10]
n bn
M
1. Show that if lim an = L and lim bn = M = 0, then lim
n
n
1
1
=
.
Mathematics 3790H Analysis I: Introduction to analysis
Trent University, Winter 2012
Solution to Assignment #4
Series business at last!
1. Show that the alternating harmonic series
(1)n+1
converges wi
Mathematics 3790H Analysis I: Introduction to analysis
Trent University, Winter 2014
Solutions to Assignment #1
Basic epsilonics
This assignment is a warm-up using something that you should have seen
Mathematics 3790H Analysis I: Introduction to analysis
Trent University, Winter 2012
Solutions to Assignment #1
A Seasonal Review
For this assignment you should probably crack open your old calculus t
Mathematics 3790H Analysis I: Introduction to analysis
Trent University, Fall 2009
Solutions to Assignment #4
A function from heck.
We rst need a bit of notation. If x is a real number, let:
cfw_x = t
Mathematics 3790H Analysis I: Introduction to analysis
Trent University, Fall 2010
Solutions to Assignment #4
The integral form of the remainder of a Taylor series
In what follows, let us suppose that
Mathematics 3790H Analysis I: Introduction to analysis
Trent University, Fall 2008
Solutions to Assignment #2
The integral form of the remainder of a Taylor series
In what follows, let us suppose that
Mathematics 3790H Analysis I: Introduction to analysis
Trent University, Fall 2008
Solutions to Assignment #3
Eeeeeeeeeeeeeeeeee!
Recall that the Taylor series at 0 of ex is 1 +
the nth remainder term
Mathematics 3790H Analysis I: Introduction to analysis
Trent University, Fall 2008
Solutions to Assignment #4
Math Trek: Dilithium? No, dilogarithm!
The dilogarithm function, Li2 (x), is usually dened
Mathematics 3790H Analysis I: Introduction to analysis
Trent University, Fall 2008
Solutions to Assignment #5
Recall that the harmonic series,
n=1
1
n
= 1+
relation, the alternating harmonic series,
n
Mathematics 3790H Analysis I: Introduction to analysis
Trent University, Fall 2008
Solutions to Assignment #1
Recall that our objective was to show that
1 1
1
1
2
=1+ + +
+ =
.
n2
4 9 16
6
n=1
1. Veri
Mathematics 3790H Analysis I: Introduction to analysis
Trent University, Fall 2008
Solutions to Assignment #6
Suppose , , and are any real numbers not in Z0 = cfw_ 0, 1, 2, . . . , and consider
the fo
Mathematics 3790H Analysis I: Introduction to analysis
Trent University, Fall 2008
Solutions to the quizzes
Quiz #1. Wednesday, 17 September, 2008. [10 minutes]
1. Find the sum of the series
n=0
Solut
Mathematics 3790H Analysis I: Introduction to analysis
Trent University, Fall 2009
Solutions to the quizzes
Quiz #1. Thursday, 24 September, 2009
The series
n=0
k
series by Sk =
n=0
(10 minutes)
1
2n
Mathematics 3790H Analysis I: Introduction to analysis
Trent University, Fall 2008
Take-home Final Exam
Due: Friday, 19 December, 2008
Instructions: Do all three of parts A C, and, if you wish, part a
Mathematics 3790H Analysis I: Introduction to analysis
Trent University, Fall 2009
Take-home Final Exam
Due on Tuesday, 22 December, 2009.
Instructions: Do all three of parts I III, and, if you wish,
Mathematics 3790H Analysis I: Introduction to analysis
Trent University, Fall 2009
Solutions to Assignment #6
Suppose , , and are any real numbers not in Z0 = cfw_ 0, 1, 2, . . . , and consider
the fo
Mathematics 3790H Analysis I: Introduction to analysis
Trent University, Fall 2009
Solution to Assignment #5
Cauchy sequences
The counterpart for sequences of the Cauchy Convergence Criterion for seri