MATH10242 Sequences and Series:
Problems 10
Please do these questions before your Tutorial in the week of May 9.
Two important changes are that:
Tutorial 1: 11am Monday May 9 is in Couland 1 Building, Pear Theatre
there are no tutorials on May 2 or May

Part II
Sequences
19
Chapter 3
Convergence
3.1
What is a sequence?
Definition 3.1.1. A sequence is a list a1 , a2 , a3 , . . . , an , . . . of real numbers labelled (or
indexed) by natural numbers. We usually write such a sequence as (an )nN , or just as

MATH10242 Sequences and Series
Toby Stafford
School of Mathematics
Alan Turing Building
Room 2.116
[email protected]
April 26, 2016
Chapter 13
Further results on power series
This chapter discusses some results that will be useful for next ye

MATH10242 Sequences and Series
Toby Stafford
School of Mathematics
Alan Turing Building
Room 2.116
[email protected]
January 25, 2016
Introduction
You probably know that
1+
1
1 1 1
+ + + + n + = 2
2 4 8
2
and even that
1
1
1 1
3
+ +
+ + n + =

MATH10242 Sequences and Series
Toby Stafford
School of Mathematics
Alan Turing Building
Room 2.116
[email protected]
April 26, 2016
Chapter 12
Power series
We now consider (the simplest case of) series where the nth term depends on a real
var

MATH10242 Sequences and Series:
Problems 9
Please do some of these questions before your Tutorial in the week of April 25.
Realistically this is too much, but working out which test to use for a given series is an important
skill, and you can only develop

MATH10242 Sequences and Series:
Problems 4
Please do these questions before your Tutorial in the week of February 29.
As always, you must ensure that you understand how to do the non-starred questions.
Question 0: This is supposed to be an easier set of q

MATH10242 Sequences and Series:
Solutions 5
Question 1 (i) limn n1000 ( 7
)n ;
8
Ans: Whenever one sees alternating terms (or even negative terms) the following a
special case of Theorem 4.1.5(ii) may be useful:
Theorem Suppose that (an )nN is a sequence

MATH10242 Sequences and Series
Toby Staord
School of Mathematics
Alan Turing Building
Room 2.116
[email protected]
April 14, 2016
Part III
Series
56
Chapter 9
Introduction to series
In this chapter we give a rigorous definition of infinite sums

MATH10242 Sequences and Series
Toby Stafford
School of Mathematics
Alan Turing Building
Room 2.116
[email protected]
April 21, 2016
10.2
The Integral Test
We consider a function f : [1, ) R which is:
(i) positive: so f (x) 0 x 0
(ii) decreasi

MATH10242 Sequences and Series
Toby Stafford
School of Mathematics
Alan Turing Building
Room 2.116
[email protected]
March 4, 2016
Chapter 6
Divergence
Example: Recall from Theorem 3.3.7 that a sequence is bounded whenever it is convergent an

MATH10242 Sequences and Series:
Solutions and Comments on the 2015 exam
A Questions:
A1. (a) A sequence (an )nN converges to a limit ` if for all > 0 there exists N N such that
|an `| < for all n N .
(b) an as n if for all real numbers K there exists N N

MATH10242 Sequences and Series:
Solutions to the 2014 exam
A Questions:
A1. A sequence (an )nN is monotone decreasing (respectively increasing) if an an+1
(respectively an an+1 ) for all n N. The Monotone Convergence Theorem states that
a monotone and bou

MATH10242 Sequences and Series:
Solutions to Exercises for Week 2 Tutorials
Question 1: Let x R. Using just the axioms for ordered fields (A09) and (Ord 1
4) from Chapter 13 of the Notes, and breaking into the cases when x is either positive,
negative or

MATH10242 Sequences and Series:
Exercises for Week 4 Tutorials
Attempt these questions before your tutorial in the week beginning 20th February.
Question 0: Some easier questions on which to get started, particularly if you find
Question 1 too hard. The s

MATH10242 Sequences and Series
Coursework Test;
17 March 2016
Fill in your details and answers on this cover sheet.
Please put away all books, notebooks, phones, etc. If you need scratch paper, we will provide it.
Name: Solution for the Exam on White pa

MATH10242 Sequences and Series
Coursework Test 12 March 2015; Solutionthe white exam
a
1
Y
2
8
3
1
2
4
N
b
c
N
N
Y
Y
d
Total Mark
N
5
Overall Mark
Answer to Qu 5:
(a) A sequence (an ) converges to a number ` if, for all > 0, there exists N N such that
|an

MATH10242 Sequences and Series:
Exercises for Week 3 Tutorials
Attempt these questions before your tutorial in the week beginning 13th February. Questions 2 and 3 are important, so make sure the spend enough time on them. As usual, an
asterisk indicates a

MATH10242 Sequences and Series: Exercises for Week 2 Tutorials
(Exercises on Real Numbers and Convergence)
Have a go at these questions before your examples classes (= tutorials) in the week starting
February 6th. (You can find your tutorial group on Blac

MATH10242 Sequences and Series
Toby Staord
School of Mathematics
Alan Turing Building
Room 2.116
[email protected]
March 1, 2016
In the last lecture we considered a number of types of sequences and their limits. In
particular we were concerned

MATH10242 Sequences and Series
Toby Stafford
School of Mathematics
Alan Turing Building
Room 2.116
[email protected]
February 17, 2016
4.2
The algebra of limits.
Hopefully you now have a feel for testing whether a given sequence is convergent

MATH10242 Sequences and Series:
Question 1: (ia)
X
cn =
n1
X xn
n1
8n
Solutions 10
.
Here the (Modified) Ratio Test gives
cn+1 xn+1 8n
|x|
=
cn xn 8n+1 = 8 .
cn+1
Thus, cn+1
<
1
|x|
<
8
(in
which
case
it
converges
absolutely)
and
cn
cn > 1 |x| >
8

MATH10242 Sequences and Series:
Solutions 1
Question 1. We break into cases. If x = 0 then x2 = 0 0 = 0 0.
If x > 0 then (Ord 4) gives x2 = x x > x 0 = 0.
Finally, by (Ord 1), the only remaining possibility is that x < 0. But, if x < 0, then
x > 0 (from 2

MATH10242 Sequences and Series:
Solutions 6
Question 1: (a) an = cos(n) n = (1)n n. Hopefully it is clear that this ought to
diverge, but it also does not tend to or to . But we should prove it formally.
None of our rules really applies directly, but what

MATH10242 Sequences and Series:
Solutions 9
P
Question 1: In each case we are clearly given
an where an = f (n) for a function f (x) that
is positive and continuous. Sometimes it is not so clear whether it is decreasing, so one needs
to say something. How

MATH10242 Sequences and Series:
Solutions 4
Question 1: In each part of the question, we first try to manipulate the expression (say
by dividing top and bottom by some function) in order to get it into a sum of terms for
which the limit is obvious. Then w

MATH10242 Sequences and Series:
Problems 8
Please do these questions before your Tutorial in the week of April 18.
X
Question 1: Use partial fractions to find:
n=1
1
.
1
4n2
Question 2: Test the series below for convergence or divergence using the tests i

MATH10242 Sequences and Series:
Problems 3
Please attempt these questions before your tutorial on the week of Feb. 22.
Question 1: Which of the following sequences converge (and to what number)? Justify
your answers. (In some of the questions you may be a

MATH10242 Sequences and Series:
Problems 6
For your Tutorial in the week of March 14, please attempt the practice test from my
website and (at the very least!) Question 1 from this sheet.
Question 1: Do the following sequences converge/diverge/tend to inf

MATH10242 Sequences and Series:
Solutions 8
Question 1: (a) Write
1
A
B
(2n 1)A + (2n + 1)B
=
+
=
.
1
2n + 1 2n 1
4n2 1
4n2
From this we get A + B = 0 and B A = 1; thus B =
t
X
n=1
t
X1
1
=
4n2 1 n=1 2
1
1
2n 1 2n + 1
The intermediate terms cancel and we