3.3
Introduction to Infinite Sequences and Series
The concept of limits and the related concepts of sequences underscores most, if not all, of the topics
which we now call ”calculus.” Here we focus directly on infinite sequences and their convergence, in
preparation for the large and extensive topic of ”infinite series.”
In short, a series is a sequence created by summing the terms of a sequence; both concepts involve
limits and grappling with the concept so central to calculus: infinity.
3.3.0
Resources
An introduction to sequences a standard part of single variable calculus. It is covered in every calculus
textbook. For example, one might look at
* Section 11.1 in Calculus, Early Transcendentals (11th ed., 2006) by Thomas, Weir, Hass, Giordano
(Pearson)
* Section 11.1 in Calculus, Early Transcendentals (6th ed., 2008) by Stewart (Cengage)
* Section 8.1 in Calculus, Early Transcendentals (1st ed., 2011) by Tan (Cengage)
* Sections 9.1 and 9.2 Calculus, Early Transcendentals (11th ed., 2009) by Anton, Bivens, Davis
(John Wiley & Sons)
* Section 10.1 in the Whitman College online textbook.
* I recommend Paul Dawkins’ calculus notes: Paul Dawkins’ calculus notes and here.
* And, of course, there is always Wikipedia: Wikipedia.
3.3.1
What is a sequence?
An infinite sequence of real numbers is merely an infinite list
a1 , a2 , a3 , ...
For example,
1. 1, 3, 5, 7, 9, ..., 2n − 1, ...
or
1
1 1 1
2. 1, , , , ..., n−1 , ...
2 4 8
2
or
1
1 1
1 1 1
1
1
3. 1, 1 + , 1 + + , 1 + + + , ..., 1 + + · + n−1 ...
2
2 4
2 4 8
2
2
4. 1, 1, 2, 3, 5, 8, 13, 21, ...
or even
5. 3, 3, 3, 3, ...
The elements in the sequence need not have a pattern – imagine a machine selecting real numbers at
random to generate the list. But in general, we will try to find a pattern; we hope that for any positive
integer n we can predict the n-th term,
Formally, a sequence {an } is a function a : N 7→ R with domain the natural numbers (N) and
codomain the real numbers (R). We will write an instead of a(n) and speak of an as the n-th term of
the sequence.
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We seek a general formula for an . Occasionally, we will be content with a recurrence relation, writing
an in terms of earlier terms. The fourth example, above, obeys the recurrence an := an−1 + an−2
The sequences in the examples above may be written as follows:
1. {an } where an := 2n − 1. (We could write this more compactly as {2n − 1}.)
2. {an } where an :=
1
,
2n−1
3. {an } where an := 1 +
1
1
+ · + n−1 = 2 − 2−n+1
2
2
4. an := an−1 + an−2 ,
and
5. {an } where an := 3.
3.3.2
Convergence of sequences
Since a sequence involves an infinite number of terms, we can ask questions about the behavior of the
sequence as the index n go to infinity. In particular, the limit of a sequence is defined to be lim an . If
n→∞
this limit exists (and is finite), we say the sequence converges to the limit. If this limit does not exist
or if it is infinite, we say the sequence diverges.
For example, with the sequence above,
1. {2n − 1} diverges to infinity.
2. {
1
2n−1
3. {1 +
1
2
} converges to 0.
+·+
1
2n−1 }
converges to 2.
4. The sequence defined by an := an−1 + an−2 diverges
and
5. {3} converges to 3.
We want to determine if sequences converge or diverge. We will calculate the limit of an as n goes to
1
infinity. For example, if the sequence is { } then
n
lim an = lim
n→∞
n→∞
1
=0
n
so the sequence converges to zero.
2n + 3
If the sequence is {
} then
n+1
2n + 3
=2
n→∞ n + 1
lim an = lim
n→∞
so the sequence converges to two.
2n2 + 3
2n2 + 3
Since lim
= ∞ the sequence {
} diverges.
n→∞ n + 1
n+1
πn
The sequence an = cos( 6 ) does not have a limit so the sequence {cos( πn
6 )} diverges.
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Convergence tools and L’hopital’s rule
There are a variety of tools we need to compute limits of sequences. One tool is L’hopital’s rule.
There are other algebraic tools.
Consider the sequence defined by
2n2 + n − 3
.
an :=
n2 + 7n
We can determine the limit in several ways.
One way is to divide top and bottom by n2 (we are looking for the highest power of n occurring in
the fraction.)
This leads to computing the limit of
2 + n1 − n32
2+0+0
2
=
= = 2.
7
n→∞
1+0
1
1+ n
lim an := lim
n→∞
Or we could use L’hopital’s rule twice on the sequence:
2n2 + n − 3
4n + 1
4
= lim
= lim
= 2.
n→∞
n→∞ 2n + 7
n→∞ 2
n2 + 7n
lim
There are other techniques. A useful technique with sequence involving powers of n is to first find the
limit of the logarithm. Once we find the limit, we need to “undo” the logarithm....
For example, suppose our sequence is an := (1 + 2/n)n . Write out the first five terms of the sequence.
Then guess its limit.
125 81 75
This is not easy. The first five terms are 3, 4,
, , .
27 16 55
If we compute the logarithm of this sequence, we have
ln(an ) = ln((1 + 2/n)n ) = n ln(1 + 2/n).
It turns out that we can use L’hopital’s rule conveniently if we write this sequence as
ln(an ) =
ln(1 + 2/n)
.
1/n
Now, as n goes to infinity, we have an indeterminate form ( 00 ) and L’hopital’s rule applies. Taking the
derivative of the numerator and denominator, we have
1
( 1+2/n
)( −2
ln(1 + 2/n)
n2 )
= lim
−1
n→∞
n→∞
1/n
n2
lim
Multiplying the denominator and numerator by n2 , we see that this is the same as
= lim
n→∞
1
( 1+2/n
)(−2)
(−1)
= 2.
So, the logarithm of our sequence converges to 2. To what does the original sequence converge? e2 !
Monotone sequences
A sequence is monotone if it does not fluctuate up and down, that is, if either it never decreases
from term to term, or it never increases from term to term.
Here are some monotone sequences:
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1. 1, 12 , 31 , 14 , 15 , ...
2. 1, 2, 3, 4, 5...
3. 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, ...
4. −1, −2, −2, −3, −3, −3, −4, −4, −4, −4, −5, −5, −5, −5, −5, ...
5. 3, 3, 3, 3, 3, 3, 3...
6. 1, 1 21 , 1 34 , 1 78 , ...
Here are some sequences which are not monotone
1. 1, − 21 , 31 , − 14 , 15 , ...
2. 1, 2, 3, 2, 3, 4, 3, 4, 5, 4...
3. {cos( πn
6 )}
A sequence is bounded if there is a real number M such that |an | ≤ M for all n.
A useful (and obvious?) result is that a bounded monotone sequence converges. For example, the
sequences below are bounded:
1. 1, 12 , 31 , 14 , 15 , ...
2. 1, 1 12 , 1 34 , 1 78 , ...
Since they are bounded and monotone, they must converge.
Comparison of sequences
Sometimes it is not easy to tell if a sequence converges, but we can compare it to another sequence
and use the comparison to advantage.
For example, consider the sequence
2 + cos(n)
an :=
3n
L’hopital’s rule will not work here since the limit lim 2 + cos(n) does not exist.
n→∞
But we can notice that since cos(n) is bounded between -1 and 1, then
2
2 + cos(n)
3
≤
≤
3n
3n
3n
2
3
We compare the sequence an with both the sequences
and
. Since both these sequences converge
3n
3n
to 0 and the sequence an is “squeezed” or “sandwiched” between those two sequences, it too must converge
to zero.
The sandwich theorem appears in the textbook by Thomas (p. 735) where there is more information
on the comparisons of sequences.
3.3.3
Sequences created by sums
A major type of sequence is that created by sums. These are called series. We will spend most of our
time on these series. Indeed, many functions are best viewed as infinite power series!
Here is a brief, first introduction to series.
Consider the sequence defined by
Sm :=
m
X
1
.
n
2
n=1
4
(1)
is defined by a sum. Its terms (partial sums) are
1
,
2
3
1 1
+ = ,
2 4
4
1 1 1
7
+ + = ,
2 4 8
8
1 1 1
1
15
+ + +
=
,
2 4 8 16
16
...
These infinite sequences defined by sums are called infinite series.
This sequence is monotone (increasing) and bounded by 1 so it converges. (Indeed, it converges to 1.)
Many of our infinite sequences, for the remainder of the course, will be defined by sums.
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