Calc 2 Lecture Notes
Section 8.2
Page 1 of 4
Section 8.2: Infinite Series
Big idea: Sometimes when you add up an infinite set of numbers, you get a finite answer. For
this to happen, most of the numbers in that set had better be pretty darn close to zero. Thus, if
the terms in a sum do not tend to zero, the sum will diverge. Remember that we are building up
to evaluating functions as:

x0 x1 x 2 x3 x 4 x5
xk
ex      
 
1 1 2 6 24 120
k 0 k !
Big skill:. You should be able to compute the sum of an infinite geometric series, and use some
basic tests to determine when a series diverges.
Definition of an Infinite Series:
Let Sn be the partial sum of the first n terms of a sequence. Then the “infinite sum” of all the
terms in the sequence is defined as follows.

n
 ak  lim  ak  lim Sn  S
n 
k 1
k 1
Practice:

1.
  0.1
k

k 1

2.
k 
k 1

3.
1
 k  k  1 
k 1
n 
Calc 2 Lecture Notes

4.
Page 2 of 4
k
1

  
k 0  3 
5. 6 12  24  48  96 

6.
Section 8.2

k
 1

  
5
k 0 
Theorem 2.1: Sum of an Infinite Geometric Series.

a
For a  0, the geometric series  ar k converges to
if r  1 and diverges if r  1 . The
1 r
k 0
number r is sometimes called the common ratio or just the ratio.
Proof:
Calc 2 Lecture Notes
Section 8.2
Page 3 of 4
Practice:
k

1
7.  3   
k 0  2 

 2 1.1
8.
k

k 0

  0.1
9.
k

k 1
Theorem 2.2: If you add up an infinite number of numbers and the sum doesn’t blow up,
then the numbers must be really small numbers.

If
a
k 1
k
converges, then lim ak  0 .
k 
kth-Term Test for Divergence. (Contrapositive of Theorem 2.2)
If lim ak  0 , then
k 

a
k 1
k
diverges.
Practice:

10. Show that the conclusion of theorem 2.2 is satisfied for
k
1

  .
k 0  3 
Calc 2 Lecture Notes
Section 8.2
Page 4 of 4
k 1
.
k
k 0

11. Determine the convergence of

12. Determine the convergence of
k.

1
k 1
Theorem 2.3: Combinations of Series (are what you think they’d be).

If
a
k 1

k
converges to A, and
b
k 1
k

converges to B, then the series
a
k 1
k
 bk   A  B , and

  ca   cA for any constant c.
k
k 1

If
a
k 1

k
diverges, and
b
k 1
k

diverges also, then the series
 a
k 1
k
 bk  diverges as well.