Calculus Of One Real Variable – By Pheng Kim Ving
Chapter 14: Infinite Series – Section 14.1: Infinite Sequences


14.1
Infinite Sequences

 

 

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1. Infinite Sequences

 

Definition Of (Infinite) Sequences

 

Finite sequences are studied in high-school algebra. A finite sequence of course is an ordered list of a finite number of
numbers. In this section we're going to discuss infinite sequences of real numbers. An infinite sequence of course is an ordered
list of infinitely many real numbers. For the sake of simplicity we'll often say “sequence” to mean “infinite sequence”.

 

Consider the following examples of sequences:

 

{1, 2, 3, 4, 5, ...},     the sequence of positive integers;
{1, 1/2, 1/3, 1/4, 1/5, ...},     the sequence of the reciprocals of the positive integers;
{1, –1, 1, –1, 1, –1, ...},     a sequence of alternating 1's and –1's.

 

The elements or terms of a sequence are listed in curly braces “{ }” and are separated by commas ",". The ellipsis “...” means
and is read “and so on”. In the second sequence, the 1st term has value 1, the 2nd term has value 1/2, the 3rd term has value
1/3, etc. The first two sequences each contain infinitely many different values. The third one contains a finite number of
different values. Values in a sequence of course may repeat themselves infinitely many times, as in the third sequence.
Although an infinite sequence may have only a finite number of different values, it's still infinite, because there are always
infinitely many terms, whether there are infinitely many values or a finite number of them. A sequence has a first term but no
last term. In general we use the word “term” to mean both the term proper and its value.

 

In the second sequence above, the 1st term is 1, the 2nd term is 1/2, the 3rd term is 1/3, ..., the nth term is 1/n, etc. The
positive integer 1 corresponds to the 1st term, the positive integer 2 corresponds to the 2nd term, ..., the positive integer n
corresponds to the nth term, etc. Each positive integer is the ordinal number of a unique term (two terms at different places
are still different terms even if they have the same value). There's a correspondence from the set of positive integers to the list
of the terms of the sequence. Clearly this correspondence is a function. The sequence is formally defined as that function.

 

Let a be that function. Then the 1st term of the sequence is a(1), denoted as a1, the 2nd term is a(2), denoted as a2, the 3rd
term is a(3), denoted a3, ..., the nth term is a(n), denoted as an, etc. The general term  is one that represents all the terms,
usually denoted an, where n can be any positive integer. The sequence {a1, a2, a3, ...} can be conveniently written as just {an}.

Let an be the general term of the 2nd sequence above. Then that sequence can be called sequence {an} and is defined by the
formula an = 1/n, meaning the nth term is 1/n, for every positive integer n. The formula an = 1/n applies to every term of the
sequence.

 

Definition 1.1 – (Infinite) Sequences

 

An infinite sequence or a sequence is a function with domain being the set N of positive integers and range being a subset of the set R of the real numbers. So a sequence is a real-valued function of the positive integer n, which usually starts from 1 up.

 

Let a be a sequence. A value of a is called a term of the sequence. For any positive integer k, the term a(k) is denoted as ak. The subscript k is the index of the term ak. The term ak is the kth term. The general term is one that represents all the terms and is denoted an, where n represents all the positive integers. If there's a formula an = f(n) for the general term, that formula applies to every term of the sequence if there's no restriction on n. The sequence a is written as {a1, a2, a3, ...} or {an}, so that:

 

{an} = {a1, a2, a3, ...},

 

where the ellipsis “...” means and is read “and so on”.

 

In practice we say and write “sequence {an}” or “sequence {a1, a2, a3, ...}”, not “sequence a”.

 

 

 

In this section when we say and write “n” we mean “positive integer n”, unless otherwise stated.

 

Three Ways To Specify A Sequence

 

The sequence {1, 1/2, 1/3, ...} is specified by listing the first few terms followed by an ellipsis. This specification is possible
because the pattern is obvious: the nth term is 1/n. This sequence can also be specified as {1/n} or an = 1/n, meaning that
the general nth term equals 1/n, a function of n. For the sequence {1, 2, 3, ...}, we notice that each term from the second
term on equals the previous term plus 1 and that the first term is 1. So this sequence can also be specified as a1 = 1, an =
an–1 + 1, n > 1 (or a1 = 1, an+1 = an + 1). For the sequence {1, 1, 2, 3, 5, 8, 13, 21, ...}, we notice that each term from the
third term on equals the sum of the previous two terms and that a1 = 1 and a2 = 1. So this sequence can also be specified as
a1 = a2 = 1, an = an–2 + an–1, n > 2 (or a1 = a2 = 1, an+2 = an + an+1).

 

Specifications Of Sequences

 

A sequence can be specified in 3 ways:

 

1. By listing the first few terms followed by an ellipsis “...”, if the pattern is obvious.
2. By providing a formula for the general term an as a function of n: an = f(n).
3. By providing a formula for the general term an as a function of some or all of the earlier terms a1, a2, ..., an–1 and
    specifying the first k terms such that the process of calculating higher terms can be done. In this case we say that
    sequence {an} is defined recursively or inductively: each term from some point on is calculated in terms of previous
    ones rather than directly as a function of n.

 

In each case it must be possible to determine any term of the sequence.

 

 

Example 1.1

 

1. List the first k terms of the following sequences followed by an ellipsis, where k is indicated for each case:

 

 

Solution
1.

 


EOS

 

Properties Of Sequences

 

The sequence {0, 0, 0, 0, 0, ...} has all its terms being 0. It's a zero sequence. The sequence {1, 2, 0, 3, 4, 0, 5, 6, 0, 7, ...}
has terms that are positive and terms that are 0. It can't be a zero sequence. It's said to be a positive sequence. The sequence
{2, 4, 6, 8, 10, ...} has all its terms being positive. It's a strictly positive  sequence. The sequence {1, –2, –3, 4, –5, 6, 0, 7, 0,
8, 0, 9, ...} has all its terms from the 6th term on being positve or 0. It's ultimately positive. It's said to be, well, ultimately
positive
. Note that the sequence {2, 0, 4, 0, 6, 0, 8, ...} is also ultimately positive; in this case, the positivity starts from the 1st
term.

The sequence:

 

 

The terms of the sequence {an} = {1, –2, 3, –4, 5, –6, 7, –8, ...} alternate in sign. It's said to be, well, alternating. We observe
that anan+1 < 0 for every n.

 

Definitions 1.2 – Properties Of Sequences

 

 

 

Remarks 1.1

 

a. Positive sequences can have terms that are 0. Not every positive sequence is strictly positive, but every strictly positive
    sequence is also positive. Similarly for negative sequences.

 

b. An increasing sequence doesn't have to always increase. It can be a constant sequence or a sequence that sometimes is
    constant and sometimes increases. A sequence that always increases is strictly increasing. Analoguously for a decreasing
    sequence. A monotonic sequence is one that's either increasing or decreasing. A sequence that sometimes increases and
    sometimes decreases isn't monotonic.

c. If U is an upper bound of a sequence then so is every real number greater than U. Similarly if L is a lower bound of a
    sequence then so is every number less than L.

 

d. Every positive sequence is also ultimately positive, but not every ultimately positive sequence is positive.

 

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2. Convergence

 

Consider the sequence:

 

 

The behavior of a converging sequence near the converging point called the limit as illustrated in Figs. 2.2 and 2.3 motivate
and justify the utilization of the verb “converge”. The terms of the sequence “converge” to the limit.

 

Fig. 2.1

 

Sequence Converging To Limit 0.

 

Fig. 2.2

 

Sequence Converging To Limit 0.

 

Fig. 2.3

 

Sequence Converging
To Limit 1.

 

 

{2.1 } Section 1.1.5.

 

Definition 2.1 – Limits And Convergence Of Sequences

 

We say that the limit of a sequence {an} is L and we write:

 

 

Note that the limit of a sequence is the limit at infinity. Observe that we say that a sequence converges iff it has a limit and
the limit is a (finite) number.

 

 

Every sequence must either converge to a (finite) number or diverge.

 

When we talk about the limit of a sequence {an}, it must be the limit at infinity, ie the limit as n approaches infinity. It makes no
sense to talk about the limit at any (finite) positive integer k, ie the limit as n approaches k, since it would require that an be
defined at all real numbers nearby k, ie that n takes on all real numbers nearby k, contrary to the definition of a sequence
requiring that n be a positive integer. That's why it's ok to say simply “the limit of {an}” and to write simply “lim an” without
specifying where n approaches, as it's understood that it must approach infinity.

 

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3. Properties Of Limits Of Sequences

 

All the standard properties of limits of functions as presented in Section 1.1.2 apply to the limits of sequences. Some of the
properties of the limits of sequences are stated below. Their proofs are similar to those for functions and thus omitted.

 

Like sum of functions, the sum of sequences {an} and {bn} is denoted {an + bn} and of course defined to be the sequence
whose nth term is the sum of the nth term of {an} and the nth term of {bn}, for every n. Similarly for other operations.

 

Properties Of Limits Of Sequences

 

If sequences {an} and {bn} both converge then:

 

 

 

In a manner analogous to the methods used for evaluating the limits of a function f(x), when evaluating the limit of a sequence
{an}, we don't use the definition of the limit to prove that a certain number is the limit of {an}, unless we're explicitly asked to
do so. Instead we use properties of the limits of sequences and/or other rules to do the evaluation.

 

Example 3.1

 

Find the limit of each of the following sequences if the sequence converges:

 

 

Solution

EOS

 

Parts b and c utilize the rule for dominating terms for rational functions. For example here's how the limit of part b is equal to
that of the dominating terms:

 

 

Example 3.2

 

Establish the convergence or divergence of each sequence:

 

 

Note

Clearly the limit in part a doesn't exist and that in part b is 0. However it's not that clear for the limit in part c. So let's reach for
a calculator and find out what it may be if it exists:

 

 

Clearly the limit should exist and be 0.5 = 1/2. In the solution we'll try to show that indeed this limit is 1/2.

 

Caution: When utilizing a calculator to find out what a limit may be, if the formula involves an angle, eg if it involves the
function sin n, then the angle n must be in radians, unless otherwise stated.

 

Solution

 

    This sequence converges to 1/2.
EOS

 

For part c, indeed our answer agrees with and confirms what our calculator suggests.

 

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4. Using Functions To Find Limits Of Sequences

 

 

Example 4.1

 

Find the limit of the sequence {n sin(3/n)} if the sequence converges.

 

Note

 

 

Solution

EOS

 

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5. The Completeness Property Of The Real-Number System

 

 

Fig. 5.1

 

The Real-Number Line.

 

 

Fig. 5.2

 

 

 

The set R of real numbers has elements that can solve all the “law-abiding” equations, ones that abide by algebraic laws for
real numbers. As a consequence it has no holes in it on the real-number line. Every point on this line corresponds to a real
number. For this reason R is said to be complete. This feature is called, well, the completeness property of the
real-number system.

 

The following statement is a formal formulation of the completeness property in terms of sequences.

 

Completeness Property

 

If a sequence is increasing or ultimately so and bounded above then it converges to some real number. Likewise if a
sequence is decreasing or ultimately so and bounded below then it converges to some real number.

 

 

The proof of this property is given in more advanced calculus or analysis courses where the real numbers are “constructed”.
It's beyond the scope of this tutorial and therefore is omitted.

 

Refer to Fig. 5.3. The above property asserts that there always exists a real number that's the limit of a sequence that's
increasing or ultimately so and bounded above, and likewise that there always exists a real number that's the limit of a
sequence that's decreasing or ultimately so and bounded below. The set R of real numbers possesses every number that's
needed. It's complete.

 

Fig. 5.3

 

{an} is ultimately increasing and is
bounded above; it converges to L.

 

Example 5.1

 

In Example 1.1 Part 3 we had the sequence {an} defined by the recursion:

 

 

Prove that {an} converges and find its limit.

 

Solution

EOS

 

 

A Note On The Irrational Numbers

 

Refer to Fig. 5.4. The rational number 1 + 1/2 = 3/2 corresponds to a point between points 1 and 2. Divide the interval [1, 2]
into 2 equal parts. The division point coincides with the point 3/2. The rational number 1 + 2/3 = 5/3 corresponds to a point
between points 1 and 2. Divide the interval [1, 2] into 3 equal parts. One of the division points coincides with the point 5/3. In

Fig. 5.4

 

No division point coincides with an
irrational number.

 

general, any rational number 1 + m/n = (m + n)/n, where m and n are positive integers and m < n, corresponds to a point
between points 1 and 2. Divide the interval [1, 2] into n equal parts. Then one of the division points coincides with the point 1 +
m/n.

 

 

This aspect is true for every irrational number, between any two consecutive integers.

 

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6. Limits Of Some Sequences

 

 

You should memorize the following limits as they're often used.

 

Limits Of Some Sequences

 

 

 

{6.1} Problem & Solution 5 Part c.

 

Proof

 

2. Let N be a positive integer such that N > |x|. For any n > N we have:

 

  

 


EOP

 

{6_2} Section 7.2 Eq. [8.1].

 

In words, Part 2 says that the exponential in n, no matter how large its base is, approaches infinity infinitely slower than the
factorial of n.

 

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7. Convergence And Boundedness

 

Clearly if a sequence {an} converges to L, then, considering the open interval (L – 1, L + 1), there exists a positive integer N
such that for all n > N, all the an's are in that interval, and so they form a bounded set. As for the previous an's (n =1, 2, ...,
N), they all are (finite) numbers, and so they form a bounded set too. Thus {an} is bounded.

 

Theorem 7.1 – Convergence Implies Boundedness

 

If a sequence converges then it's bounded.

 

 

Proof
Let {an} be a sequence that converges. Let L be its limit. Then there exists a positive integer N such that for all n > N we have
|Lan| < 1, ie L – 1 < an < L + 1, so that |an| < |L| + 1. Let K = max(|a1|, |a2|, ..., |aN|, |L| + 1). Then |an| is less than
or equal to K for all n. Thus {an} is bounded.

EOP

 

The converse is false. Boundedness doesn't imply convergence. For example, {1/n} is bounded and converges, while {(­–1)n} is
bounded but doesn't converge.

 

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Problems & Solutions


1.
 

Solution

 

 

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Solution

 

 

 

Note

 

For part b, examination of two groups of terms already shows that the sequence simply diverges, so there's no need to
examine other groups if any, in this case there's another group, which consists of the (4n)th terms, for all n > 0.

 

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3. Let the sequence {an} be defined recursively by:

 

  

 

   Prove that {an} is convergent and find its limit. Hint: Use the completeness property to prove convergence.

Solution

 

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4. Is the statement “If {an} and {bn} both diverge, then {anbn} diverges.” true or false? If it's true, prove it, or if it's false, give a
    counter-example.

Solution

 

It's false. Let {an} = {(–1)n} = {–1, 1, –1, 1, –1, 1,  . . .} and {bn} = {(–1)n+1} = {1, –1, 1, –1, 1, –1,  . . .}. Then {an} and {bn}
both diverge. But {anbn} = {(–1)n(–1)n+1} = {(–1)2n+1} = {–1, –1, –1, –1, –1, –1,  . . .} converges to –1.

 

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c. {xn}, where |x| < 1, converges to 0. See also Limits Of Some Sequences Part 1.

 

Solution

 

 

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