Absolute vs.
Conditional Convergence
Alternating Series
and the
Alternating Series Test
Series with Positive Terms
Recall that series in which all the terms are positive
have an especially simple structure when it comes to
convergence.
Because each term that is added is positive, the
sequence of partial sums is increasing.
So one of two things happens:
1. The partial sums stay bounded and the
series converges,
OR
2. The partial sums go off to infinity and the
series diverges.
What happens when a Series has
some terms that are negative?
There are several Possibilities:
•All the terms are negative.


n 0
•Finitely many terms are negative.
3
n8
1 
1 1 1 1
   
2 4 8 16
•Infinitely many negative terms and infinitely many positive terms.
1 
1 1 1 1
   
2 3 4 5
1 1 1 1 1
1      
2 4 8 16 32
Alternating Series
Definition: An alternating series is one whose terms
alternate in sign. For a sequence (cn) of positive numbers,
there are two possibilities:
c0 - c1+ c2 - c3+c4 . . . Or -c0+c1- c2 + c3 - c4 . . .
In some ways, this situation is the most conducive to
convergence, since the positive and negative terms have a
tendency to cancel each other out, thus preventing the partial
sums from getting too large.
Note: the Nth term test for divergence still applies.
Consider: -1+1-1+1-1+1. . .
One of the most important Examples is:
The Alternating Harmonic Series
1
1 1 1 1 1 1
     
2 3 4 5 6 7
The alternating harmonic
In order to determine whether the series series
converges,
seemswe
to converge to
need to examine the partial sums of the series.a point about here
Look at
Example 1 on
pg. 576 of OZ.
1
1/2
1 2 3 4 5 6 7 8 9 10
This suggests
The Alternating Series Test
Theorem: (Alternating Series Test) Consider the series
c1 - c2 + c3 - c4 . . . and -c1+ c2 - c3+ c4 . . .
Where
lim ck  0
c1 > c2 > c3 > c4 > . . .> 0 and
n 
Then the series converge, and each sum S lies between
any two successive partial sums.
That is, for all k, either
S k  S  S k 1
or
S k 1  S  S k ,
depending on whether k is even or odd.
The Idea behind the AST
We have already seen the crucial picture.
a0
S7
S is between S7 and S8
and | S 7  S8 |  | a8 |
S
a0-a1
Note:
S8
1 2 3 4 5 6 7 8 9 10
The error estimate given at the end of the theorem is also
obvious from the picture.
The Idea behind the AST
We have already seen the crucial picture.
a0
S7
S
a0-a1
S8
1 2 3 4 5 6 7 8 9 10
There were two things that made this picture “go.”
• The size (in absolute value) of the terms was decreasing.
• The terms were going to zero.
Absolute and Conditional
Convergence
Definition: Let  an be any series
• If  an converges, then  an is said to converge absolutely.
• If  an diverges but  an converges, then  a is said to
converge conditionally.
n
Fact: Any series that converges absolutely, also converges
in the partial sum sense.
In this case, the absolute sum of the series is greater than
or equal to the sum of the series.
The converse is not true. It is possible for a series to
converge but not to converge absolutely.
Quintessential example: the alternating harmonic series.
The Alternating Series Test (revised)
Theorem: (Alternating Series Test) Consider the
series c1 - c2 + c3 - c4 . . . and -c1+c2 - c3 + c4 . . .
Where
lim ck  0
c1 > c2 > c3 > c4 > . . .> 0 and
n 
Then the series converge, and each sum S lies between
any two successive partial sums.
In particular, for all k, the error in approximation is
S  Sk  Sk 1  Sk  ck 1.