Calculus Of One Real Variable – By Pheng Kim Ving


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1. Infinite Sequences 
Definition Of
(Infinite) Sequences
Finite sequences are studied in highschool 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 a_{1}, the 2nd term is a(2),
denoted as a_{2}, the 3rd
term is a(3), denoted a_{3}, ..., the nth term is a(n), denoted as a_{n},
etc. The general term is one that represents
all the terms,
usually denoted a_{n}, where n can be any positive integer. The sequence {a_{1}, a_{2}, a_{3}, ...} can be conveniently written as just {a_{n}}.
Let a_{n}
be the general term of the 2nd sequence above. Then that sequence can be called
sequence {a_{n}} and is defined by the
formula a_{n} = 1/n, meaning the nth
term is 1/n, for every positive integer n. The formula a_{n}
= 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 realvalued 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 a_{k}. The subscript k is the index of the term a_{k}. The term a_{k} is the kth term. The general term is one that represents all the terms and is denoted a_{n}, where n represents all the positive integers. If there's a formula a_{n} = 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 {a_{1}, a_{2}, a_{3}, ...} or {a_{n}}, so that: {a_{n}} = {a_{1}, a_{2}, a_{3}, ...}, where the ellipsis “...” means and is read “and so on”. In practice we say and write “sequence {a_{n}}” or “sequence {a_{1}, a_{2}, a_{3}, ...}”, 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 a_{n}
= 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 a_{1} = 1, a_{n} =
a_{n}_{–1}
+ 1, n > 1 (or a_{1} = 1, a_{n}_{+1} = a_{n} + 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 a_{1} = 1 and a_{2} = 1. So this
sequence can also be specified as
a_{1} = a_{2} = 1, a_{n} = a_{n}_{–2} + a_{n}_{–1}, n > 2 (or a_{1} = a_{2} = 1, a_{n}_{+2} = a_{n} + a_{n}_{+1}).
A sequence can be specified in 3 ways: 1. By listing the first few terms followed by an
ellipsis “...”, if the pattern is obvious. 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 {a_{n}}
= {1, –2, 3, –4, 5, –6, 7, –8, ...} alternate in sign. It's said to be, well, alternating.
We observe
that a_{n}a_{n}_{+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 
^{{2.1 }} Section 1.1.5.
Definition 2.1 – Limits And Convergence Of Sequences
We say that the limit of a sequence {a_{n}} 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 
Every sequence must either converge to a (finite) number or diverge.
When we talk about the limit of a sequence {a_{n}}, 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 a_{n} 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 {a_{n}}”
and to write simply “lim a_{n}”
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 {a_{n}} and {b_{n}} is denoted {a_{n} + b_{n}}
and of course defined to be the sequence
whose nth term is the sum of the nth term of {a_{n}}
and the nth term of {b_{n}},
for every n. Similarly for other operations.
Properties Of Limits Of Sequences
If sequences {a_{n}} and {b_{n}} 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
{a_{n}}, we don't use the
definition of the limit to prove that a certain number is the limit of {a_{n}}, 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 RealNumber System 

Fig. 5.1 The RealNumber Line. 

Fig. 5.2 
The set R of real numbers has elements
that can solve all the “lawabiding” equations, ones that abide by algebraic
laws for
real numbers. As a consequence it has no holes in it on the realnumber 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
realnumber 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 
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 {a_{n}}
is ultimately increasing and is 
Example 5.1
In Example 1.1 Part 3 we had the sequence {a_{n}} defined by the recursion:
Prove that {a_{n}} 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 
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 {a_{n}}
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 a_{n}'s
are in that interval, and so they form a bounded set. As for the previous a_{n}'s (n =1, 2, ...,
N), they all are (finite)
numbers, and so they form a bounded set too. Thus {a_{n}}
is bounded.
Theorem 7.1 – Convergence
Implies Boundedness
If a sequence converges then it's bounded. 
Proof
Let {a_{n}} 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
L – a_{n} < 1, ie L – 1 < a_{n} < L + 1, so that a_{n} < L + 1. Let K
= max(a_{1},
a_{2},
..., a_{N}, L + 1). Then a_{n} is less than
or equal to K for all n. Thus {a_{n}}
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.
Problems & Solutions 
1.
Solution
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.
3. Let the sequence {a_{n}}
be defined recursively by:
Prove that {a_{n}} is convergent and
find its limit. Hint: Use the completeness property to prove convergence.
Solution
4. Is the statement “If {a_{n}}
and {b_{n}} both diverge, then {a_{n}b_{n}}
diverges.” true or false? If it's true, prove it, or if it's false, give a
counterexample.
Solution
It's false. Let {a_{n}}
= {(–1)^{n}} = {–1, 1, –1, 1, –1, 1, . . .} and {b_{n}}
= {(–1)^{n}^{+1}} = {1, –1, 1, –1, 1, –1, . . .}. Then {a_{n}}
and {b_{n}}
both diverge. But {a_{n}b_{n}} = {(–1)^{n}(–1)^{n}^{+1}} = {(–1)^{2}^{n}^{+1}} = {–1, –1,
–1, –1, –1, –1, . . .} converges to –1.
c. {x^{n}}, where x < 1, converges to 0. See also Limits Of Some Sequences Part 1.
Solution
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