Download Infinite Series - Madison College

Calc 2 Lecture Notes
Section 8.2
Page 1 of 2
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.
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:

n
 ak  lim  ak  lim Sn  S
n 
k 1
k 1
n 
Practice:

  0.1

k
k 1

k 
k 1

1
 k  k  1 
k 1
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.
Practice:
k

1
3  

k 0  2 

 2 1.1
k

k 0

  0.1
k 1
k

Calc 2 Lecture Notes
Section 8.2
Page 2 of 2
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:

k 1


k
k 0
Theorem 2.3: Combinations of Series (are what you think they’d be).

If
 ak converges to A, and
k 1

 bk converges to B, then the series
k 1

a
k 1
k
 bk   A  B , and

  ca   cA for any constant c.
k
k 1

If
 ak diverges, and
k 1

 bk diverges also, then the series
k 1

 a
k 1
k
 bk  diverges as well.