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MATH10242 Sequences and Series
We shall prove the second and third of these formulas in this course unit, but the
first one is too difficult and will be done in your lectures on real and complex analysis in
the second year. Here, “real analysis” means the study of functions from real numbers
to real numbers, from the point of view of developing a rigorous foundation for calculus
(differentiation and integration) and for other infinite processes.2 The study of sequences
2and “complex analysis” refers to the complex numbers, not (necessarily) complexity in the sense of
“complicated”!
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and series is the first step in this programme.
This also means there are two contrasting aspects to this course. On the one hand we
will develop the machinery to produce formulas like the ones above. On the other hand
it is also crucial to understand the theory that lies behind that machinery. This rigorous
approach forms the second aspect of the course—and is in turn the first step in providing
a solid foundation for real analysis.
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Chapter 1
Before We Begin
1.1 Some Reminders about Mathematical Notation
1.1.1 Special sets
We use the following notation throughout the course.
R - the set of real numbers;
R+ - the set of strictly positive real numbers, i.e. R+ = {x ∈ R : x > 0};
Q - the set of rational numbers;
Z - the set of integers (positive, negative and 0);
N (or Z+) - the set of natural numbers, or positive integers {x ∈ Z : x > 0}. (In this
course, we do not count 0 as a natural number. We can use some other notation like Z≥0
for the set of integers greater than or equal to 0.)
∅ - the empty set.
1.1.2 Set theory notation
The expression “x ∈ X” means x is an element (or member) of the set X. For sets A,
B, we write A ⊆ B to mean that A is a subset of B (i.e. every element of A is also an
element of B). Thus ∅ ⊆ N ⊆ Z ⊆ Q ⊆ R.
Standard intervals in R: if a, b ∈ R with a ≤ b, then
• (a, b) ={x ∈ R : a < x < b}; • [a, b) ={x ∈ R : a ≤ x < b};
• (a, b] ={x ∈ R : a < x ≤ b}; • [a, b] ={x ∈ R : a ≤ x ≤ b};
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• (a,∞) ={x ∈ R : a < x}; • [a,∞) ={x ∈ R : a ≤ x};
• (−∞, b) ={x ∈ R : x < b}; • (−∞, b] ={x ∈ R : x ≤ b};
• (−∞,∞) = R.
1.1.3 Logical notation
The expression “∀x ...” means “for all x ...” and “∃x ...” means “there exists at least one
x such that ...”. These are usually used in the context “∀x ∈ A ...” meaning “for all
elements x of the set A ...”, and “∃x ∈ A ...” meaning “there exists at least one element
x in the set A such that ...”.
Thus, for example, “∀x ∈ R x > 1” means “for all real numbers x, x is greater than
1” (which happens to be false) and “∃x ∈ R x > 1” means “there exists a real number x
such that x is greater than 1” (which happens to be true).
1.1.4 Greek letters
The two most commonly used Greek letters in this course are δ (delta) and ε (epsilon).
They are reserved exclusively for (usually small) positive real numbers.
Others are α (alpha), β (beta), γ (gamma), λ (lambda), θ (theta-usually an angle),
η (eta) and Σ (capital sigma - the summation sign which will be used when we come to
study series in Part II).
1.1.5 Where we’re headed and some things we’ll see on the way
In Part I we aim to understand the behaviour of infinite sequences of real numbers,
meaning what happens to the terms as we go further and further on in the sequence. Do
the terms all gradually get as close as we like to a limiting value (then the sequence is said
to converge to that value) or not? The “conceptual” aim here is to really understand what
this means. To do that, we have to be precise and avoid some plausible but misleading
ideas. It’s worthwhile trying to develop, and refine, your own “pictures” of what’s going
on. We also have to understand the precise definition well enough to be able to use it
when we calculate examples, though we will gradually build up a stock of general results
(the “Algebra of Limits”), general techniques and particular cases, so that we don’t have
to think so hard when faced with the next example.
Part II is about “infinite sums” of real numbers: how we can make a sensible definition
of that vague idea and then how we can calculate the value of an infinite sum - if it exists.
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We also need to be able to tell whether a particular “infinite sum” does or doesn’t make
sense/exist. Sequences appear here in two ways: first as the sequence of numbers to be
“added up” (and the order of adding up does matter, as we shall see); second as a crucial
ingredient in the actual definition of an “infinite sum” (“infinite series” is the official
term). What we actually do is add up just the first n terms of such an infinite series -
call this value the n-th partial sum - and then see what happens to this sequence (note)
of partial sums as n gets bigger and bigger. If that sequence of partial sums converges to
a limit then that limit is what we define to be the sum of the infinite series. Hopefully
that makes sense to you and seems like it should be the right definition to use. Anyway,
it works and, again, we have the conceptual aspect to get to grips with as well as various
techniques that we can (try to) use in computations of examples.
Here are some of the things we prove about our concept of limit: a sequence can have
at most one limit; if a sequence is increasing but never gets beyond a certain value, then
it has a limit; if a sequence is squeezed between two other sequences which have the same
limit l, then it has limit l. These properties help clarify the concept and are frequently
used in arguments and calculations. We also show arithmetic properties like: if we have
two sequences, each with a limit, and produce a new sequence by adding corresponding
terms, then this new sequence has a limit which is, as you might expect, the sum of the
limits of the two sequences we started with.
Then we turn to methods of calculating limits. We compare standard functions (poly-
nomials, logs, exponentials, factorials, ...) according to how quickly they grow, but ac-
cording to a very coarse measure - their “order” of growth, rather than rate of growth
(i.e. derivative). That lets us see which functions in a complicated expression for the n-th
term of a sequence are most important in calculating the limit of the sequence. There
will be lots of examples, so that you can gain some facility in computing limits, and there
are various helpful results, L’Hoˆpital’s Rule being particularly useful.
While the properties of sequences are, at least once you’ve absorbed the concept,
quite natural, infinite series hold quite a few surprises and really illustrate the need
to be careful about definitions in mathematics (many mathematicians made errors by
mishandling infinite series, especially before the concepts were properly worked out in
the 1800s). Given an infinite series, there are two questions: does it have a sum? (then
we say that it “converges”, meaning that the sequence of partial sums has a limit - the
value of that infinite sum) and, if so, what is the value of the sum? There are a few series
(e.g. a geometric series with ratio < 1) where we can quite easily compute the value but,
in general this is hard. It is considerably easier to determine whether a series has a sum
or not by comparing it with a series we already know about. Indeed, the main test for
convergence that we will use, the Ratio Test, is basically comparison with a geometric
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series.
Many infinite series that turn up naturally are “alternating”, meaning that the terms
are alternately positive and negative. So, in contrast with the corresponding series where
all the terms are made positive, there’s more chance of an alternating sequence converging,
because the terms partly cancel each other out. Indeed, remarkably, it’s certain provided
the absolute values of the individual terms shrink monotonically to 0!
We’ll finish with power series: infinite series where each term involves some variable
x - you could think of these as “infinite polynomials in x”. Whether or not a power
series converges (i.e. has a sum), depends very much on the value of x, convergence being
more likely for smaller values of x. In fact, the picture is as good as it could be: there’s
a certain “radius of convergence” R (which could be 0 or ∞ or anything in between,
depending on the series) such that, within that radius we have convergence, outside we
have divergence, and on the boundary (x = ±R) it could go either way for each boundary
point (so we have to do some more work there).
1.1.6 Basic properties of the real numbers
It will be assumed that you are familiar with the elementary properties of N, Z and Q
that were covered last semester in MATH10101/10111. These include, in particular, the
basic facts about the arithmetic of the integers and a familiarity with the Principle of
Mathematical Induction. One may then proceed to construct the set R of real numbers.
There are many ways of doing this which, remarkably, all turn out to be equivalent in a
sense that can be made mathematically precise. One method with which you should be
familiar is to use infinite decimal expansions as described in Section 13.3 of [PJE].
Here we extract some of the basic arithmetic and order properties of the real numbers.
First, just as for the set of rational numbers, R is a field. That means that it satisfies
the following conditions.
(A0) ∀a, b ∈ R one can form the sum a+ b and the product a · b (also written as just ab).
We have that a+ b ∈ R and a · b ∈ R; (existence of two binary operations.)
Properties of addition.
(A1) ∀a, b, c ∈ R, a+ (b+ c) = (a+ b) + c (associativity of +);
(A2) ∀a, b ∈ R, a+ b = b+ a (commutativity of +);
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(A3) ∃0 ∈ R,∀a ∈ R, a+ 0 = 0 + a = a (additive identity);
(A4) ∀a ∈ R, there exists an element in R (denoted −a) such
that a+ (−a) = 0 = (−a) + a; (additive inverse);
Properties of multiplication.
(A5) ∀a, b, c ∈ R, a · (b · c) = (a · b) · c (associativity of ·);
(A6) ∀a, b ∈ R, a · b = b · a (commutativity of ·);
(A7) ∃1 ∈ R,∀a ∈ R, a · 1 = 1 · a = a (multiplicative identity);
(A8) ∀a ∈ R, if a 6= 0 then there exists an element in R (denoted a−1
or 1/a) such that a · a−1 = 1 = a−1 · a (multiplicative inverse);
Combining the two operations.
(A9) ∀a, b, c ∈ R, a · (b+ c) = a · b+ a · c (the distributive law).
These axioms (A0)-(A9) list the basic arithmetic/algebraic properties that hold in
R and from which all the other such properties may be deduced.
Importantly,
Example A The identities and inverses, both additive and multiplicative, are unique.
Solution If 0 and 0′ are two additive identities then
0 = 0 + 0′ since 0′ is an additive identity
= 0′ since 0 is an additive identity.
Thus 0 = 0′, the identity is unique.
If a has two additive inverses, b and c then 0 = a+ b since b is the additive inverse of
a. Adding c to both sides gives
c+ 0 = c+ (a+ b).
On the left use that 0 is the additive identity and on the right use associativity, so
c = (c+ a) + b.
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Then c is the additive inverse of a so
c = 0 + b.
Finally, 0 is the additive identity so c = b, and the inverse is unique.
I leave the proofs for the multiplicative identities and inverses to the students.
This uniqueness of identities and inverses is important. It will used in some of the
following proofs. Consider 0, the additive identity; what can be said of the multiplication
0 · x for x ∈ R? Similarly, −1 is the additive inverse of 1, so what can be said of the
multiplication (−1) ·x? The only axiom that combines addition and multiplication is the
distributive law and so it should be no surprise it is used within the proof of
Example B For all x ∈ R, 0 · x = 0 and (−1) · x = −x.
Solution Given x ∈ R,
0 · x = x · 0 (A6)
= x · (0 + 0) (A3)
= x · 0 + x · 0 (A9)
= 0 · x+ 0 · x (A6).
Add −(0 · x) to both sides.
0 = 0 · x+ (−(0 · x))
= (0 · x+ 0 · x) + (−(0 · x)) (A1)
= 0 · x+ (0 · x+ (−(0 · x))) (A4)
= 0 · x+ 0 (A3)
= 0 · x.
Next
x+ (−1) · x = 1 · x+ (−1) · x (A7)
= x · 1 + x · (−1) (A6)
= x · (1 +−1) (A9)
= x · 0 (A4)
= 0 · x (A6)
= 0,
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by the first result of this example. This shows that (−1) · x is the additive inverse of x.
Yet, by Example A, the additive inverse of an element is unique. Thus (−1) ·x = −x.
The second result here can be generalised, for all x, y ∈ R,
(−x) · y = −(x · y) = x · (−y).
What about inverses of inverses?
Example C For all x ∈ R, −(−x) = x and, provided x 6= 0, (x−1)−1 = x.
Solution Left to student.
Or sums and products of additive inverses?
Example D For all x, y ∈ R, (−x) + (−y) = −(x+ y) and (−x) · (−y) = x · y.
Solution Left to student.
And a couple of often used results are
Theorem (Cancellation Law for Addition) If a, b and c ∈ R and a+ c = b+ c, then
a = b.
Theorem (Cancellation Law for Multiplication) If a, b and c ∈ R, c 6= 0 and ac = bc
then a = b.
Proofs Left to student.
For repeated multiplication we use the usual notation: for a ∈ R, a2 is an abbreviation
for a · a (similarly a3 is an abbreviation for a · (a · a), etc.).
Example 1.1.1. Prove the identity (x+ y)2 = x2 + 2xy + y2 above.
Solution We have
(x+ y)2 = (x+ y)(x+ y) (by definition),
= (x+ y) · x+ (x+ y) · y (by A9),
= x · (x+ y) + y · (x+ y) (by A6),
= x · x+ x · y + y · x+ y · y (by A9),
= x · x+ x · y + x · y + y · y (by A6),
= x · x+ 1 · (x · y) + 1 · (x · y) + y · y (by A7),
= x · x+ (1 + 1) · (x · y) + y · y (by A9 and A6),
= x · x+ 2 · (x · y) + y · y (by definition),
= x2 + 2xy + y2 (by definition).
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Actually we have also used A1 many times here. This allowed us to ignore brackets
in expressions involving many + symbols.
Finally, for now, we can define two further binary operations on R:
Subtraction For all x, y ∈ R, x− y = x+ (−y),
Division For all x, y ∈ R, y 6= 0,
x÷ y
(
=
x
y
= x/y
)
= x · y−1.
Example For all s, t, x, y ∈ R with x, y 6= 0 we have
• (s− t)2 = s2 − 2st+ t2,
• (s− t)(s+ t) = s2 − t2,
• (s/x)(t/y) = (st)/(xy),
• (s/x) + (t/y) = (sy + tx)/(xy).
Solution Left to student.
Note that if we replace R in these axioms by Q then they still hold; that is, Q is also
a field (but Z is not since it fails (A8)). The point of giving a name (“field”) to an “arith-
metic system” where these hold is that many more examples appear in mathematics, so it
proved to be worth isolating these properties and investigating their general implications.
Other fields that you will have encountered by now are the complex numbers and the
integers modulo 5 (more generally, modulo any prime).
The real and rational numbers have some further properties not shared by, for exam-
ple, the complex numbers or integers modulo 5.
Namely, the real numbers form an ordered field. This means that we have a total
order relation < on R. All the properties of this relation and how it interacts with the
arithmetic operations follow from the following axioms:
(Ord 1) ∀a, b ∈ R, exactly one of the following is true: a < b or b < a or a = b;
(trichotomy)
(Ord 2) ∀a, b, c ∈ R, if a < b and b < c, then a < c; (transitivity)
(Ord 3) ∀a, b ∈ R, if a < b then ∀c ∈ R, a+ c < b+ c;
(Ord 4) ∀a, b ∈ R, if a < b then ∀d ∈ R+, a · d < b · d.
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As usual, we write a ≤ b to mean either a = b or a < b.
You do not need to remember exactly which rules have been written down here. The
important point is just that this short list of properties is all that one needs in order to
be able to prove all the other standard facts about <. For example, we have:
Example 1.1.2. Let us prove that if x is positive (i.e. 0 < x) then −x is negative
(i.e. −x < 0). So suppose that 0 < x. Then by (Ord 3), 0 + (−x) < x + (−x).
Simplifying we get −x = 0 + (−x) < x+ (−x) = 0, as required.
Other facts that follow from these properties include:
∀x ∈ R x2 ≥ 0,
∀x, y ∈ R x ≤ y ⇔ −x ≥ −y, et cetera.
It left as an exercise to prove these facts using just the axioms (Ord 1–4). Also see
the first exercise sheet.
However, there are more subtle facts about the real numbers that cannot be deduced
from the axioms discussed so far. For example, consider the theorem that
√
2 is irrational.
This really contains two statements: first that
√
2 6∈ Q (that is, no rational number
squares to 2; this was proved in MATH10101/10111); second that there really is such a
number in R - there is a (positive) solution to the equation X2 − 2 = 0 in R. And this
definitely is an extra property of the real numbers—simply because it not true for the
rational numbers (which satisfies all the algebraic and order axioms that we listed above).
So, we need to formulate a property of R that expresses that there are no “missing
numbers” (like the “missing number” in Q where
√
2 should be). Of course, we have
to say what “numbers” should be there, in order to make sense of saying that some of
there are “missing”. The example of
√
2 might suggest that we should have “enough”
numbers so as to be able to take n-th roots of positive numbers and perhaps to solve
other polynomial equations and following that idea does lead to another field - the field
of real algebraic numbers - but we have a much stronger condition (“completeness”) in
mind here. We will introduce it in Chapter 2, just before we need it.
1.1.7 The Integer Part (or ‘Floor’) Function
From any of the standard constructions of the real numbers one has the fact that any
real number is sandwiched between two successive integers in the following precise sense:
∀x ∈ R, ∃n ∈ Z such that n ≤ x < n+ 1.
(The proof of the existence of n requires the Completeness axiom, discussed later.) The
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integer n that appears here, the greatest integer less than or equal to x, is unique and
is denoted [x]; it is called the integer part of x. The function x 7→ [x] is called the
integer part function. Note that 0 ≤ x− [x] < 1.
Example 1.1.3. [1.47] = 1, [pi] = 3, [−1.47] = [−2].
Note Many people denote the greatest integer less than or equal to x by bxc, the floor
function. In a similar manner, we can talk of the least integer, greater than or equal to
x, denoted by dxe, the ceiling function.
We will often want to choose an integer larger than a given real number x. Instead
of using the ceiling function we will choose [x] + 1; this is not always the least integer
greater than or equal to x (consider when x is an integer) but it suffices for our needs.
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Part I
Sequences
16
Chapter 2
Convergence
2.1 What is a Sequence?
Definition 2.1.1. 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)n∈N, or as (an)n≥1
or just as (an). We say that an is the nth term of the sequence.
The word “sequence” suggests time, with the numbers occurring in temporal sequence:
first a1, then a2, et cetera. Indeed, some sequences arise this way, for instance as successive
approximations to some quantity we want to compute. Formally, a sequence is simply a
function f : N→ R, where we write an for f(n).
We shall be interested in the long term behaviour of sequences, i.e. the behaviour of
the numbers an when n is very large; in particular, do the approximations converge on
some value?
Example 2.1.2. Consider the sequence 1, 4, 9, 16, . . . n2, . . .. Here, the nth term is n2.
So we write the sequence as (n2)n≥1 or (n2)n≥1 or just (n2). What is the nth term of the
sequence 4, 9, 16, 25, . . .? What are the first few terms of the sequence
(
(n− 1)2)
n≥1?
Example 2.1.3. Consider the sequence 2, 3/2, 4/3, 5/4, . . . . Here, an = (n+ 1)/n. The
sequence is ((n+ 1)/n)n≥1.
Example 2.1.4. Consider the sequence −1, 1,−1, 1,−1, . . .. A precise and succinct way
of writing it is
(
(−1)n)
n≥1. The nth term is 1 if n is even and −1 if n is odd.
Example 2.1.5. Consider the sequence ((−1)n/3n)n≥1. The 5th term, for example, is
−1/243.
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Example 2.1.6. Sometimes we might not have a precise formula for the nth term but
rather a rule for generating the sequence, E.g. consider the sequence 1, 1, 2, 3, 5, 8, 13, . . .,
which is specified by the rule a1 = a2 = 1 and, for n ≥ 3, an = an−1 + an−2. (This is the
Fibonacci sequence.)
Long term behaviour: In Examples 2.1.2 and 2.1.6 the terms get huge, with no bound
on their size (we shall say that they tend to ∞).
However, for 2.1.3, the 100th term is
101
100
= 1 +
1
100
,
the 1000th term is
1001
1000
= 1 +
1
1000
.
It looks as though the terms are getting closer and closer to 1. (Later we shall express
this by saying that ((n+ 1)/n)n≥1 converges to 1.)
In Example 2.1.4 , the terms alternate between −1 and 1 so don’t converge to a single
value.
In Example 2.1.5, the terms alternate between being positive and negative, but they
are also getting very small in absolute value (i.e. in their size when we ignore the minus
sign): so ((−1)n/3n)n≥1 converges to 0.
Before giving the precise definition of “convergence” of a sequence, we require some
technical properties of the modulus (i.e. absolute value) function.
We now make the convention that unless otherwise stated, all variables (x, y,
l, ε, δ ...) range over the set R of real numbers.
2.2 The Triangle Inequality
We define the modulus, |x| of x, also called the absolute value of x.
|x| =
x if x ≥ 0,−x if x < 0.
Note that |x| = max{x,−x}, so x ≤ |x| and −x ≤ |x|.
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Theorem 2.2.1 (The Triangle Inequality). For all x, y, we have
|x+ y| ≤ |x|+ |y|.
Proof. We have x ≤ |x| and y ≤ |y|. Adding, we get x+ y ≤ |x|+ |y|.
Also, −x ≤ |x| and −y ≤ |y|, so −(x+ y) = (−x) + (−y) ≤ |x|+ |y|.
It follows that max{x+ y,−(x+ y)} ≤ |x|+ |y|, i.e. |x+ y| ≤ |x|+ |y|, as required.
Remark: For x, y ∈ R, |x−y| is the distance from x to y along the “line” R. The triangle
inequality is saying that the sum of the lengths of any two sides of a triangle is at least
as big as the length of the third side, as is made explicit in the following corollary. (Of
course we are dealing here with rather degenerate triangles: the name really comes from
the fact that in this form, the triangle inequality is also true for points in the plane.)
Corollary 2.2.2 (Also called the Triangle Inequality). For all a, b, c, we have
|a− c| ≤ |a− b|+ |b− c|.
Proof.
|a− c| = |(a− b) + (b− c)| ≤ |a− b|+ |b− c|
by Theorem 2.2.1 with x = (a− b) and y = (b− c)).
Lemma 2.2.3. Some further properties of the modulus function:
(a) ∀x, y |x · y| = |x| · |y| and, if y 6= 0,∣∣∣∣xy
∣∣∣∣ = |x||y| ;
(b) ∀x, y |x− y| ≥ ∣∣|x| − |y|∣∣;
(c) ∀x, l and ∀ε > 0, |x− l| < ε ⇐⇒ l − ε < x < l + ε.
Proof. Exercises: (a) - consider the cases; (b), (c) - see the Exercise Sheet for Week 3.
Now we come to the definition of what it means for a sequence (an)n≥1 to converge
to a real number l. We want a precise mathematical way of saying that “as n gets bigger
and bigger, an gets closer and closer to l” (in the sense the distance between an and l
tends towards 0).
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2.3 The Definition of Convergence
Definition 2.3.1. We say that a sequence (an)n≥1 converges to the real number l if
the following holds:
∀ε > 0 ∃N ∈ N such that for all n ∈ N with n ≥ N we have |an − l| < ε.
Equivalently,
∀ε > 0,∃N ∈ N : ∀n ≥ 1, n ≥ N =⇒ |an − l| < ε.
We use various expressions and notations for when this holds:
• an tends to l as n tends to infinity, written an → l as n→∞.
• the limit of (an)n≥1 as n tends to infinity equals l, written limn→∞ an = l.
Example 2.3.2. Consider the sequence(
n+ 1
n
)
n≥1
of 2.1.3.
Claim: (n+ 1)/n→ 1 as n→∞.
Proof Let ε > 0 be given. The definition requires us to show that there is a natural
number N such that for all n ≥ N , ∣∣∣∣n+ 1n − 1
∣∣∣∣ < ε.
So we look for N so that ∀n ≥ N , |(1 + 1/n)− 1| < ε, which is equivalent to requiring
that ∀n ≥ N , |1/n| < ε.
The N here will depend on ε (in fact in this example, as in most, no choice of N will
work for all ε, so any choice of value for N will be in terms of ε). Since the last inequality
is equivalent to ∀n ≥ N , 1/ε < n we take N to be any natural number greater than 1/ε,
say for definiteness [ε−1] + 1.
Note i. Any N > 1/ε would have sufficed but, when you are required to show something
exists, I would recommend you exhibit an explicit example, such as [ε−1] + 1.
ii. The N depends inversely on ε; as ε gets smaller, the N gets larger. This is what we
would expect, if we want to terms of the sequence to be closer to the limit value then we
will have to look further along the sequence. Use this as a sanity check on your answers.
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Example 2.3.3. Now consider the sequence(
(−1)n
3n
)
n≥1
of 2.1.5. The claim is that (−1)n/3n → 0 as n→∞.
Proof: Let ε > 0 be given. To prove the claim we must find N ∈ N so that ∀n ≥ N ,∣∣∣∣(−1)n3n − 0
∣∣∣∣ < ε.
In fact N = [ε−1] + 1 (which implies 1/N < ε) works here. For suppose that n ≥ N .
Then ∣∣∣∣(−1)n3n − 0
∣∣∣∣ = ∣∣∣∣(−1)n3n
∣∣∣∣ = |(−1)n||3n|
by Lemma 2.2.3(a)
=
1
3n
≤ 1
n
since it is easy to show (by induction) that ∀n ∈ N, 3n ≥ n. But 1/n ≤ 1/N < ε, and we
are done.
Note You could finish this example differently;∣∣∣∣(−1)n3n − 0
∣∣∣∣ = 13n ≤ 13N < ε
as long as 3N > 1/ε, i.e. N > log3 (1/ε). This choose N = [log3 (1/ε)]+1. Though we will
use the logarithm function in this course it will not be defined properly until next year,
thus I recommend keeping away from it if at all possible. Also, I would claim [ε−1] + 1
is ‘simpler’ than [log3 (1/ε)] + 1 thus I would further recommend continuing and giving
‘simple’ upper bounds for |an − l|, e.g. here I consider 1/N to be simpler than 1/3N ; that
it is larger is immaterial, all we require is that the final bound can be made smaller than
ε by taking N sufficiently large.
The definition of convergence is notoriously difficult for students to take in, and rather
few grasp it straight away, especially to the extent of being able to apply it. So here’s
a different, but equivalent, way of saying the same thing. Maybe one of the definitions
might make more sense to you than the other - it can be useful to look at something from
different angles to understand it. And, of course, the more examples you do, the more
quickly you will get the picture.
By Lemma 2.2.3(c) the condition |an − l| < ε is equivalent to saying that l − ε <
21
an < l + ε. This in turn is equivalent to saying that an ∈ (l − ε, l + ε). So, to say that
an → l as n → ∞ is saying that no matter how small an interval we take around l, the
terms of the sequence (an)n≥1 will eventually lie in it, meaning that, from some point on,
every term lies in that interval.
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