Download Infinite Sequences

In…nite Sequences
Dr. Philippe B. Laval
Kennesaw State University
October 9, 2008
Abstract
This hand out is an introduction to in…nite sequences. It gives the
main de…nitions and presents some elementary results.
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1.1
In…nite Sequences (8.1)
Elementary Concepts
Simply speaking, a sequence is simply a list of numbers, written in a de…nite
order:
fa1 ; a2 ; a3 ; :::an ; an+1 ; :::g
where the elements ai represent numbers. In this section, we only concentrate
on in…nite sequences. Here are some general facts about sequences studied in
this class:
Every number an in the sequence has a successor an+1 ; the sequence never
stops since we study in…nite sequences.
Think of the index of a particular element as indicating the position of
the element in the list.
The index can also be associated with a formula when the elements in the
list are generated by one (see below). As such, the index of the numbers
does not have to start at 1, though it does most of the time.
If we call n0 the value of the starting index, then there is a number an
for every n n0 . Thus, we can de…ne a function f such that an = f (n)
where n is a natural number.
In this class, we will concentrate on in…nite sequences of real numbers. A
more formal de…nition of a sequence is as follows:
De…nition 1 (sequence) A sequence of real numbers is a real-valued function
f whose domain is a subset of the non-negative integers, that is a set of the
form fn0 ; n0 + 1; :::g where n0 is an integer such that n0
0.. The numbers
an = f (n) are called the terms of the sequence.
1
1
The typical notation for a sequence is (an ), or fan g or fan gn=1 where an
denotes the general term of the sequence.
1
Remark When a sequence is given by a function ff (n)gn=n0 , the function f
must be de…ned for every n n0 .
A sequence can be given di¤erent ways
The elements of the sequence are given.
A formula to generate the terms of the sequence is given.
A recursive formula to generate the terms of the sequence is given.
Example 2 The examples below illustrate sequences given by listing all the elements (it is really not possible since these are in…nite sequences),
1. f1; 5; 10; 4; 98; 1000; 0; 2; :::g.
2.
1 1
1; ; ; ::: . Though the elements are listed, we can also guess a formula
2 3
to generate them, what is it?
3. f 1; 1; 1; 1; 1; 1; :::g. Though the elements are listed, we can also guess
a formula to generate them, what is it?
Example 3 The examples below illustrate sequences given by a simple formula.
Notice that n does not have to start at 1.
1.
1
n
1
. The elements of the sequence are:
n=1
term of the sequence is an =
2.
1 1
1; ; ; ::: . The general
2 3
1
.
n
p p p
1
n 3 n=3 . The elements are 0; 1; 2; 3; ::: . In this case, n could
not start at 1.
p
n 1
3. f( 1) gn=2 . The elements of the sequence are f1; 1; 1; 1; :::g.
Example 4 The examples below illustrate sequences given by a recursive formula.
1.
a1 = 1
. We can use this formula to generate all the terms.
an = 2an 1 + 5
But the terms have to be generated in order. For example, in order to get
a10 , we need to know a9 , and so on. Using the formula, we get that
a2
=
=
2
2a1 + 5
7
Having found a2 , we can now generate a3
a3
=
=
2a2 + 5
19
And so on.
2. A special sequence generated recursively is the Fibonacci
8 sequence, named
f1 = 1
<
f2 = 1
after an Italian mathematician. It is de…ned as follows:
.
:
fn = fn 1 + fn 2
We can use this formula to generate the terms of the sequence. This sequence was devised to model rabbit population. fn represents the number
of pairs of rabbits after n months, assuming that each month, a pair of
rabbit produces a new pair which becomes productive at age 2 months.
Example 5 Other sequences
1. Arithmetic sequence. A sequence is arithmetic if the di¤ erence between
two consecutive terms is constant. Let r be this constant. Then, we have
a1
a2
an
a0
a1
an
1
= r =) a1 = a0 + r
= r =) a2 = a1 + r = a0 + 2r
:::
= r =) an = an 1 + r = a0 + nr
Thus, there is a de…ning formula for arithmetic sequences. For example
an arithmetic sequence starting at 2, such that the di¤ erence between two
consecutive terms is 3 is de…ned by: an = 3n + 2
2. Geometric sequence. A sequence is geometric if the ratio of two consecutive terms is constant. Let q denote this constant. Then, we have
a1
a0
a2
a1
an
an 1
= q =) a1 = qa0
= q =) a2 = qa1 = q 2 a0
:::
= q =) an = qan 1 = q n a0
Thus, there is a de…ning formula for geometric sequences. For example a
geometric sequence starting at 2, such that the ratio between two consecutive terms is 3 is de…ned by: an = 2 3n .
Sequences can be plotted. However, the plot of a sequence will consists of
dots, since they are only de…ned at the integers. Figures 1, 2, 3, and 4 show
some sequences being plotted.
3
Figure 1: Plot of an =
1.2
1
n
Limit of a Sequence
Given a sequence fan g, one of the questions we try to answer is: what is the
behavior of an as n ! 1? Is an getting closer and closer to a number? In other
words, we want to …nd lim an .
n!1
De…nition 6 (limit of a sequence) A sequence fan g converges to a number
L as n goes to in…nity if an can be made as close as one wants to L, simply
by taking n large enough. In this case, we write lim an = L. If lim an = L
n!1
n!1
and L is a …nite number, we say that fan g converges. Otherwise, it diverges.
Sometimes, we will make the distinction between diverges to in…nity and simply
diverges. In the …rst case, we still know what the sequence is doing, it is getting
large without bounds.
A sequence may diverge for several reasons. Its general term could get
arbitrarily large (go to in…nity), as shown on …gure 3. Its general term could
also oscillate between di¤erent values without ever getting close to anything.
n
n
This is the case of f( 1) g, or f( 1) ln ng as shown on …gure 4.
4
Figure 2: Plot of an =
sin n
n
Another way to understand this is that if lim an = L then jan Lj goes to 0
n!1
when n ! 1. It should also be noted that if lim an = L, then lim an+1 = L.
n!1
n!1
In fact, lim an+p = L for any positive integer p.
Graphically, the meaning of lim an = L is as follows. Consider the sequence
shown on …gure 5 whose plot is represented by the dots. The sequence appears
to have 3 as its limit, this is indicated by the horizontal solid line through 3. If
we draw a region having the line y = 3 at its center, then saying that lim an = 3
means that there exists a certain value of n (denote it n0 ) such that if n > n0 ,
then all the dots corresponding to the plot of the sequence will fall in the region.
On …gure 5 , we drew two regions, one with dotted lines, the other one with
dashdot lines. We can see that in both cases, after a while, the sequence always
falls in the region. Of course, the more narrow the region is, the larger n0 will
be. In other words, if we want to guarantee that an is closer to 3, we have
to look at an for larger values of n. It appears that for the larger region, the
sequence falls in the region if n > 10. For the smaller region, it happens when
n > 22 (approximately).
5
Figure 3: Plot of ln n
Let us …rst state, but not prove, an important theorem.
Theorem 7 If a sequence converges, its limit is unique.
We now look at various techniques used when computing the limit of a
sequence. As noticed above, a sequence can be given di¤erent ways. How its
limit is computed depends on the way a sequence is given.
We begin with the easiest case, one we are already familiar with from Calculus I. If the sequence is given by a function, we can, in many instances, use
our knowledge of …nding the limit of a function, to …nd the limit of a sequence.
This is what the next theorem tells us.
Theorem 8 If lim f (x) = L and an = f (n) then lim an = L
x!1
n!1
This theorem simply says that if we know the function which generates the
general term of the sequence, and that function converges as x ! 1 then the
sequence converges to the same limit. We know how to do the latter from
Calculus I.
6
n
Figure 4: Plot of ( 1) ln n
n
.
n+1
x
x
The function generating this sequence is f (x) =
. Since lim
=1
x!1 x + 1
x+1
n
(l’Hôpital’s rule), lim
= 1.
n+1
Example 9 Find lim
Be careful, this theorem only gives a de…nite answer if lim f (x) = L. If
x!1
the function diverges, we cannot conclude. For example, consider the sequence
an = cos 2n . The function f such that an = f (n) is f (x) = cos 2 x. This
function diverges as x ! 1. Yet, an = cos 2n = 1 for any n. So, lim an = 1.
Theorem 10 If an
lim bn = L
bn
cn for n
n0 and lim an = lim cn = L then
n!1
n!1
n!1
This is the equivalent of the squeeze theorem. You will notice in the statement of the theorem that the condition an bn cn does not have to be true
for every n. It simply has to be true from some point on.
Example 11 Find lim
n!
nn
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Figure 5: Limit of a sequence
Since n! is only de…ned for integers, we cannot …nd a function f such that
an = f (n). Thus, we cannot use the previous theorem. We use the squeeze
theorem instead. We notice that:
0
=
n!
nn
1 2
n n
1
n
3
n
By the squeeze theorem, it follows that lim
:::
:::
n
n
n!
= 0.
nn
Theorem 12 If lim jan j = 0 then lim an = 0
n!1
n!1
This is an application of the squeeze theorem using the fact that jan j
an
jan j. This theorem is often useful when the general term of a sequence
n
contains ( 1) as suggests the next example.
8
n
( 1)
.
n
n
( 1)
1
First, we notice that jan j =
=
which converges to 0 by the previous
n
n
example. Hence, by theorem 12, an ! 0 also.
Example 13 Find lim an for an =
We now look at a theorem which is very important. Unlike the other theorems, we can prove this one as it is not very di¢ cult.
Theorem 14 The sequence fxn g converges to 0 if jxj < 1. It converges to 1 if
x = 1. It diverges otherwise.
Proof. We consider several cases.
case 1: x = 1. Then, xn = 1n = 1. Thus, the sequence converges to 1.
n
1. Then, xn = ( 1) which diverges.
case 2: x =
case 3: x = 0. Then, xn = 0n = 0. Thus, the sequence converges to 0.
n
case 4: jxj < 1 and x 6= 0. Then, ln jxj < 0. Thus, ln jxj = n ln jxj !
n
1. Thus, jxj ! 0 as n ! 1. Hence, by the previous theorem, xn ! 0
as n ! 1.
case 5: x > 1. Then, xn ! 1.
case 6: x < 1. xn oscillate between positive and negative values, which
are getting larger in absolute value. Thus it also diverges.
Finally, we state a theorem which is the equivalent for sequences of the limit
rules for functions.
Theorem 15 Suppose that fan g and fbn g converge, and that c is a constant.
Then:
1.
2.
3.
4.
5.
6.
lim (an + bn ) = lim an + lim bn
n!1
n!1
lim (an
bn ) = lim an
n!1
n!1
lim (an bn ) =
lim an
n!1
lim
n!1
an
bn
=
n!1
lim an
n!1
lim bn
n!1
n!1
lim bn
n!1
lim bn
n!1
providing lim bn 6= 0
n!1
lim can = c lim an
n!1
n!1
lim c = c
n!1
7. lim apn =
n!1
h
lim an
n!1
ip
if p > 0 and an > 0.
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Since we only consider limits as n ! 1, we will omit it and simply write
lim an . The way we …nd lim an depends greatly on how the sequence is given.
1
.
n
1
In this case, the function which generates the terms of the sequence is f (x) = .
x
1
Since lim
= 0, it follows that lim an = 0 by theorem 8.
x!1 x
Example 16 Find lim an for an =
Example 17 Find lim ln
therefore lim ln
n
n+1
n
n
. From a previous example, lim
= 1,
n+1
n+1
= ln 1 = 0.
n
Example 18 Find lim ( 1) .
Since the terms of this sequence are f 1; 1; 1; 1; :::g, they oscillate but never
get close to anything. The sequence diverges. In contrast, the sequence of the
…rst example also oscillated. But it also got closer and closer to 0.
sin n
Example 19 Find lim
.
n
Though we can …nd a function to express the general term of this sequence,
since sin x diverges, we cannot use theorem 8 to try to compute the limit of this
sequence. We note that
1 sin n 1
Therefore
1
n
sin n
n
1
n
sin n
1
=
Since lim = 0, by the squeeze theorem for sequences, it follows that lim
n
n
0.
ln n
n
ln x
The function de…ning the general term is f (x) =
. Since
x
Example 20 Find lim
ln x
lim
x!1 x
It follows that lim
1
= lim x by l’Hôpital’s rule
x!1 1
1
= lim
x!1 x
= 0
ln n
= 0.
n
10
Example 21 Assuming that the sequence given recursively by
(
an+1
a1 = 2
1
= (an + 6)
2
converges, …nd its limit.
Let L = lim an . If we take the limit on both sides of the relation de…ning the
sequence, we have
lim an+1
=
lim
1
(an + 6)
2
1
(L + 6)
2
2L = L + 6
L = 6
L =
So, lim an = 6.
1.3
Increasing, Decreasing and Bounded Sequences
De…nition 22 (increasing, decreasing sequences) A sequence fan g is said
to be
1. increasing if and only if an < an+1 for each nonnegative integer n.
2. non-decreasing if and only if an
an+1 for each nonnegative integer n.
3. decreasing if and only if an > an+1 for each nonnegative integer n.
4. non-increasing if and only if an
an+1 for each nonnegative integer n.
5. monotonic if any of these four properties holds.
To show that a sequence is increasing, we can try one of the following:
1. Show that an < an+1 for all n.
2. Show that an
an+1 < 0 for all n.
3. If an > 0 for all n, then show that
an
< 1 for all n.
an+1
4. If f (n) = an , show that f 0 (x) > 0
5. By induction.
Example 23 Let an =
n
. Show fan g is increasing.
n+1
11
Method 1: look at
an
an+1
an
an+1
=
=
=
<
Method 2: Let f (x) =
n
n+1
n+1
n+2
n (n + 2)
(n + 1) (n + 1)
n2 + 2n
n2 + 2n + 1
1
x
1
. Then f 0 (x) =
2 >0
x+1
(x + 1)
De…nition 24 (bounded sequences) A sequence fan g is said to be bounded
from above if there exists a number M such that an M for all n. M is called
an upper bound of the sequence. A sequence fan g is said to be bounded from
below if there exists a number m such that an
m for all n. m is called a
lower bound of the sequence. A sequence is bounded if it is bounded from above
and below.
Example 25 Consider the sequence an = cos n. Since 1 cos n 1, an is
bounded from above by 1 and bounded from below by 1. So, an is bounded.
1
1
1
. We have 0 <
n n=1
n
sequence is bounded below by 0 and above by 1.
Example 26 Consider the sequence
1. Thus the
Remark 27 Clearly, if M is an upper bound of a sequence, then any number
larger than M is also an upper bound. So, if a sequence is bounded from above,
it has in…nitely many upper bounds. Similarly, if a sequence is bounded below
by m, then any number less than m is also a lower bound.
Theorem 28 A sequence which is increasing and bounded from above must
converge.
Theorem 29 A sequence which is decreasing and bounded from below must
converge.
We use this theorem to prove that certain sequences have a limit. Before
we do this, let us review induction. This concept is often needed to show a
sequence is increasing or bounded.
Theorem 30 (Induction) Let P (n) denote a statement about natural numbers with the following properties:
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1. The statement is true when n = 1 i.e. P (1) is true. This is call the base
case.
2. P (k + 1) is true whenever P (k) is true for any integer k.
Then, P (n) is true for all n 2 N.
1
Example 31 Prove that the sequence de…ned by a1 = 2 and an+1 = (an + 6)
2
is increasing and bounded. Find its limit.
Increasing: We show by induction that an+1 > an .
1
1
(a1 + 6) = (2 + 6) = 4 > a1 .
2
2
– Assume the result is true for any integer k, that is assume ak+1 > ak .
Show the result is true for k + 1 that is ak+1 > ak+1 .
– Base case. a2 =
ak+2
1
(ak+1 + 6)
2
1
>
(ak + 6) since ak+1 > ak
2
= ak+1
=
– The result follows by induction.
Bounded: We show by induction that an
– Base case: a1 = 2
6.
6.
– Assume the result is true for any integer k, that is ak
result is also true for k + 1 that is ak+1 6.
ak
=)
=)
6 =) ak + 6
1
(ak + 6) 6
2
ak+1 6
6, show the
12
– The result follows by induction.
Limit: We have already computed the limit of such a sequence. Now that
we know the limit exists since fan g in increasing and bounded above, the
idea is to give it a name, say lim an = L and …nd what L is using the limit
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rules as follows:
an+1
=
()
()
()
()
()
1
1
(an + 6) () lim (an+1 ) = lim (an + 6)
2
2
1
L = lim (an + 6)
2
1
L = (lim an + lim 6)
2
1
L = (L + 6)
2
2L = L + 6
L=6
So
lim an = 6
We …nish with an important remark.
Remark 32 Consider a sequence fan g.
If fan g has a limit, it must be bounded. Try to explain why.
Not every bounded sequence has a limit. Give an example of a bounded
sequence with no limit.
1.4
Things to Know and Problems Assigned
Be able to write the terms of a sequence no matter which way the sequence
is presented.
Be able to tell if a sequence converges or diverges. If it converges, be ale
to …nd its limit.
Be able to tell if a sequence is increasing or decreasing.
Related problems assigned:
1. on pages 565, 566 # 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 35,
39, 41, 43
2. Let fan g be a sequence such that an = f (n) for some function f .
Suppose that lim f (x) does not exist. Does it mean that fan g
x!1
diverge? Explain or give a counter example.
3. Explain why if a sequence has a limit then it must be bounded.
4. Give examples of bounded sequences which do not have limits.
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