Calc 2 Lecture Notes
Section 8.1
Page 1 of 12
Chapter 8: Infinite Series
The whole point of this chapter is to be able to write any given function as an infinite sum
of power function terms, with each consecutive term resulting in a better approximation to the
given function. This is an extension of the linear and quadratic approximations to a function.
The following sequence of graphs shows successively better approximations to the exponential
function.
“Constant” Approximation
Cubic Approximation








y = e^x











y = 1 + x + x^2 / 2 + x^3
/ 6


y = 1





Linear Approximation




















Fifth-Order Polynomial Approximation

y = 1+x+x^2/2+x^3/6+x^4/24+x^5/120


y = e^x

y = e^x















y = 1 + x









Quadratic Approximation
y = 1 + x + x^2 / 2


y = 1 + x + x^2/2 + x^3/6
+ x^4/24


y = e^x

y = e^x



Quartic Approximation





y = e^x









Calc 2 Lecture Notes
Section 8.1
Page 2 of 12
The table below shows the range of x values for which each approximation is good to within
10% error.
Approximation for ex
ex  1
ex  1  x
1 2
x
2
1
1
e x  1  x  x 2  x3
2
6
1
1
1 4
e x  1  x  x 2  x3 
x
2
6
24
1
1
1 4
1 5
e x  1  x  x 2  x3 
x 
x
2
6
24
120
ex  1  x 
Range of x values for which the
approximation is good to 10% error
-0.09 < x < 0.10
-0.39 < x < 0.53
-0.70 < x < 1.10
-1.01 < x < 1.74
-1.31 < x < 2.43
-1.61 < x < 3.15
Clearly, the last approximation is the best approximation.
If we wanted a better approximation, we could obtain it by noticing the pattern to the
1
1
1 4
1 5
1
x 
x   xk 
coefficients: e x  1  x  x 2  x 3 
2
6
24
120
k!
Thus, we can write an “exact” polynomial for the exponential function using an infinite number

xk
of terms: e x  
k 0 k !
In this chapter, we will build up to figuring out formulas like the one above by first
learning techniques for analyzing infinite series and determining a range of values over
which they converge. This will give us a basis for understanding how to write any function
as an infinite series of power terms, as in the example of the exponential function above.
Calc 2 Lecture Notes
Section 8.1
Page 3 of 12
Section 8.1: Sequences of Real Numbers
Big idea: The limit of a sequence of real numbers can be determined in much the same way that
the limit of a function can be determined.
Big skill: You should be able to find the limit of a sequence of real numbers given the “general
term.”
Definition of a Sequence:
A sequence is any function whose domain is the set of integers starting with some integer n0
1
(usually, n0 = 0 or n0 = 1). For instance, the function a  n   for n = 1, 2, 3, … defines the
n
sequence
1 1 1 1
1, , , , , .
2 3 4 5
1
Each number in the sequence is called a term. a  n   is called the general term, since it
n
gives a general formula for calculating any term. Also, from now on, we’ll use the subscript
notation for a sequence: an  a  n  .


1 
1 1 1 1 
Set notation for all the terms in a sequence: an n1      , , , ,  .
 n n1 1 2 3 4 
Practice:
n
1. Compute and graph the first few terms of an 
.
n 1
Calc 2 Lecture Notes
Section 8.1
Page 4 of 12
To graph a sequence on the TI-83:
(i). Set the graphing mode to Seq using the MODE button.
(ii). Enter the “general term” using the Y= button.
(iii). Set the window properties using the WINDOW button.
(iv). Graph the sequence using the GRAPH button.
To graph a sequence in Winplot (see picture on next page):
(i). Choose Recursive… from the Equa menu.
(ii). Enter recursive formulas for the new x and new y values of the graph.
The new x will always be x + 1 for this chapter.
The new y will be the general term, but you must use (x + 1) in place of n.
(iii). Choose Sequence… from the One menu.
(iv). Set the initial x value to 1 and the initial y value to the value of the first term in the
sequence, then click on the draw button.
Calc 2 Lecture Notes
Section 8.1
Page 5 of 12
Calc 2 Lecture Notes
Section 8.1
Page 6 of 12
Definition 1.1: Convergence / Divergence of a Sequence.

The set an n  n converges to L if and only if given any number  > 0 there is an integer N for
0
which an  L   for every n > N. If there is no such number L, then we say the sequence
diverges.
(This definition is the starting point for proving Theorems 1.1, 1.2, and 1.3…)
Practice:

 n 
2. Prove that the sequence 
 converges to 1.
 n  1 n 1
Theorem 1.1: Limits of Combinations of Sequences.


If an n  n and bn n  n both converge, then
0
0
(i).
lim  an  bn   lim an  lim bn
(ii).
lim  an  bn   lim an  lim bn
(iii).
lim  anbn   lim an
(iv).
 a  lim an
lim  n   n
(assuming lim bn  0 ).
n 
n  b
lim
b
 n  n n
n
n
n
n
n 
n 
n 

Practice:
n 
 lim b 
n 
n
2n  5
. (Trick: divide top and bottom by the smallest leading term in the
n  3n  2
rational expression, or do long division.)
3. Evaluate lim
Calc 2 Lecture Notes
Section 8.1
Page 7 of 12
2n 2  1
. (Trick: divide top and bottom by the smallest leading term in the
n  5n  3
rational expression, or do long division.)
4. Evaluate lim
Theorem 1.2: The limit of a sequence is the limit of its function.
If x  and lim f  x   L , then lim f  n   L for n .
x 
n
Note that the converse is not true. Counterexample: lim cos  2 n 
n 
Practice:
n2
.
n  e n
5. Evaluate lim
Theorem 1.3: Squeeze Theorem


Suppose that an n  n and bn n  n both converge to the limit L. If there is an integer n1  n0 such
0
0
that all n  n1 guarantees that an  cn  bn, then cn n  n converges to L as well.

0
Practice:
sin  n 
 0 using the squeeze theorem.
n 
n
6. Prove lim
Calc 2 Lecture Notes
Section 8.1
Page 8 of 12
Corollary 1.1 to the Squeeze Theorem:
If lim an  0 , then lim an  0 as well.
n 
n 
Practice:
7. Prove
 1
lim
n 
n2
n
 0 using corollary 1.1.
Definition 1.2: The factorial.
For any integer n > 1, the factorial of n, written as n!, is defined as the product of the first n
positive integers.
n!  1 2  3   n
We define the factorial of zero to be one: 0!  1.
Practice:
8. Compute 10!
9. Compute
7!
3! 7  3!
10. Write the product of even numbers from 2 to 2n using the factorial.
Calc 2 Lecture Notes
Section 8.1
Page 9 of 12
11. Write the product of odd numbers from 1 to 2n + 1 using the factorial.
2n
12. Evaluate lim .
n  n !
10n
.
n  n !
13. Evaluate lim
14. Notice in the previous two examples that the factorial always “overwhelms” the
exponential function, no matter how large the base of the exponential function. State

xk
why this is important in the formula for the exponential function series: e x  
k 0 k !
Calc 2 Lecture Notes
Section 8.1
Page 10 of 12
Definition 1.3: Increasing and Decreasing sequences.

The sequence an n 1 is increasing if a1  a2  a3   an  an1 
The sequence an n 1 is decreasing if a1  a2  a3   an  an1 
If a sequence is either increasing or decreasing it is called monotonic.

Key trick: To determine whether a series is monotonic, look at the ratio of successive terms
an 1
. If the ratio of a term to the previous term is always greater than 1, then the sequence is
an
increasing. If that ratio is always less than 1, then the sequence is decreasing.
Practice:
15. Determine whether an 
n
is increasing, decreasing, or neither.
n 1
16. Determine whether an 
n!
is increasing, decreasing, or neither.
en
Calc 2 Lecture Notes
Section 8.1
Page 11 of 12
Definition 1.4: Bounded sequences.

The sequence an n 1 is bounded if there is a number M > 0 (called a bound) for which |an| < M
for all n.
Practice:

 3  4n 2 
17. Show that  2
 is bounded.
 n  1 n 1
Theorem 1.4: Convergence of bounded monotonic sequences.
Every bounded, monotonic sequence converges.
Common sense: If a sequence is bounded, it never exceeds a certain range of values. If the
sequence is decreasing, then it is approaching some value inside that range, or the value of –M
(like in the picture above). If the sequence is increasing, it approaches some value inside that
range, or the value of +M.
Note: theorem 1.4 does not tell us the limit of the sequence; just the fact that the sequence
converges. Sometimes just knowing a sequence converges is more important than knowing the
limit, anyway.
Calc 2 Lecture Notes
Section 8.1
Page 12 of 12
Practice:

 n! 
18. Prove that  n  converges.
 n n1
The Completeness Axiom.
If a nonempty set S of real numbers has a lower bound, then it has a greatest lower bound.
Equivalently, if S has an upper bound, then it has a least upper bound.
Note that this is an axiom; it a statement we accept as being true without proof. Axioms form the
foundation upon which all other theorems are based.
The completeness axiom, when combined with the definition of the convergence of a sequence
allows us to prove theorem 1.4 (every bounded monotonic sequence converges).