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 n 0,
x2
x3
xn
x
+
+ +
+ Rn (x) ,
ex = 1 + +
1!
2!
3!
n
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 eld of
a
dierential geometry. Along the way he came up w
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 function such that f (n) (x) is dened
and continuous for
Mathematics 3790H Analysis I: Introduction to analysis
Trent University, Fall 2010
Solutions to Assignment #3
A slice of
1. Verify that
n=0
16n2
1
converges absolutely. [4]
+ 16n + 3
Solution. Observe that the terms
16n2 n2 for n 1, we have
16n2
n=1
are
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) = lim
n
1+
1 1
1
+ + + ln(n)
2 3
n
n
1
Since ln(n) = 1 x d
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
for all x R.
1. Work out the power series for ax , whe
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 diverges. [4]
np
n=1
ln(n)
converges when p > 1, but does not
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 nd its limit. [10]
1. Suppose we dene a sequence an as f
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, part as well.
Show all your work. You may use your tex
Mathematics 3790H Analysis I: Introduction to analysis
Trent University, Winter 2012
Quiz Solutions
Quiz #1. Monday, 16 Thursday, 19 January, 2012. [10 minutes]
1. Suppose X R has sup(X) = lub(X) = a. Show that if Y = cfw_ x | x X , then
inf(Y ) = glb(Y )
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 N and n > 0 = 0. Use this fact to help
prove the Archi
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 areas
n
n=1
1
and comparing it to the area under the g
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
=
.
n bn
M
Hint: This is easier if you rst show that if lim
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 without using the
n
n=1
Alternating Series Test. [5]
Solu
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 some version
of in rst-year caculus, the denition of li
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 textbook and
skim through the material on sequences and
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 = the distance from x to the nearest integer
= min cfw_x x
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 c is a real number and f (x) is a function such
that f
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 a is a real number and f (x) is a function such
that f
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 at 0, i.e.
x
1!
+
x2
2!
+
x3
3!
+ . Let Rn,0 (x) denot
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 as the sum of an innite series:
Li2 (x) =
x2
x3
x4
xn
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=1
1
2
+
(1)n+1
n
1
3
+
1
4
=1
+ , diverges, while its
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. Verify the following trigonometric identity. (So long as x
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 following power series:
1+
( + 1) ( + 1) 2 ( + 1)( + 2) (
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
Solution. The series
n=0
2 n
3
2 n
3
= 1+
term 1 and common
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
= 1 + 1 + 1 + 1 + sums to 2. Denote the kth partial sum
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 as well. Show all
your work. You may use your textbooks
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, part as well. Show all
the work necessary to support yo
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 following power series:
1+
( + 1) ( + 1) 2 ( + 1)( + 2) (
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 series is the
following notion:
Definition. A sequence a0 ,