BC 3
Sequences and Series 2
Name: ____________________
Definition: An infinite series is an expression of the form

a
k 1
k
 a1  a2  a3 
Also called an infinite sum or simply a series, we are basically adding infinitely many numbers
together. You’ve been doing this for some time now. Remember,


1
k
1
 
MI-4 & Geometric Series:     2  1 .
1
k 1  2 
1
2
3
3
3
1

  .
Fourth grade: 0 .33333  
10 100 1000
3

In calculus, we are more thoughtful and define this idea carefully and logically.
Some important terminology: Given the infinite series

a
k 1
k
 a1  a2  a3 
, we define:
n
The nth partial sum, Sn by: Sn   ak  a1  a2  a3 
k 1
The nth remainder, or nth tail by: Rn 

a
k  n 1
k
an .
 an 1  an  2  an  3 
The sequence of partial sums: S n n 1 .

The sequence an n 1 is called the sequence of general terms.


Note:
i.
a
k 1
k
 S n  Rn
ii. Sn  Sn1  an for n  1

Definition: We say the series
a
k 1
k
converges if and only if the sequence of partial sums
converges. In the case lim  Sn   S we say the infinite series converges to S and write
n 
If

a
k 1
k
does not converge, we say it diverges.
IMSA BC3
Sequences and Series 2.1

a
k 1
k
S.
Determine whether each of the following series converges by finding a simple formula for the
partial sum S n and evaluating lim  Sn  .
n 

(1)
1
 (n  1)(n  1)
n2
(2)

k 1
(3)
1
3

k
  
n 1
n 1
n
e  x dx 

This last problem suggests an intimate connection between infinite series and improper integrals,
which we will explore more fully after a short digression into the theory of convergent and
divergent series.
IMSA BC3
Sequences and Series 2.2
Theorem (the nth term test): If lim an  0 , then
n 

a
n 1
n
diverges.
Proof: Left to the astute student.
Determine (with explanation) whether the following series converge or diverge. If it converges,
state the limit.

(1)
 1

1  
n
n 1 
(2)
  3

1
n
n 1

n
1 

4n 

(3) Find a formula for the nth partial sum of
 n 
 ln  n  1 .
n 1
diverge?

(4)
1
n
(This is called the harmonic series)
n 1
(5) Let’s look at the series

1
 k!.
k 0
IMSA BC3
Sequences and Series 2.3
Does the series converge or
i. Explain why Sn  is increasing.
k
1
ii. Show that S n  1     for each n  0 . Does this imply Sn  is bounded?
k 0  2 
n

1
iii. 

k  n 1 k !

k
1

  = _________________?
k  n 1  2 
iv. Use your calculator or computer to find

1
 k ! to the nearest thousandth. Explain
k 0
carefully how you know your answer is within .001
One way to look at the previous problem is as follows:

Suppose that
 an converges and
n 1

n
n 1
k 1
 an  S . Since S   ak 

a
k  n 1
k
 S n  Rn , we can say that
Rn is the error when using S n to approximate S. We write S  Sn  Rn . So we often want to
find a bound on Rn .
IMSA BC3
Sequences and Series 2.4