8.6 Alternating Series, Absolute and Conditional
Convergence

n 1

1

an
Alternating series have the form 
n 1
a1 – a2 + a3 – a4 + ….

Alternating Series Test

n 1

1

an converges if:

n 1
1. an > 0 for all n
2. an ≥ an+1 for all n ≥ N for some integer N
lim an  0
3. n



Ex.
 1
n 1
n 1
1
n

Ex.
 1
n 1
n 1
2n 2
4n 2  1

Alternating Series Estimation Theorem

n 1

1

an
If the alternating series 
satisfies the three
n 1
conditions of the Alternating Series Test, then for all n ≥ N
n
sn   1 ai  a1  a2  a3  ...  1
i1

i1
n 1
an
approximates the sum L of the series with an error whose
absolute value is less than an+1, the value of the first unused
term. The remainder L – sn has the same sign as the first
unused term.

Ex.
 1
n 1
n 1
n
n3  1

1. Estimate the error in using the sum of the first four terms to
approximate the sum of the entire series.
2. Approximate the sum so the error has magnitude ≤ .01.
Do: 1. Are the following series convergent or divergent?

 1
a.
n 1

n 1

c.
1
n10

n 1


 1
d.
n 1
n 1

1
10 n
1
10 n

b.


n 1
1
10 n

1
 n10n
n1
e.


f.
1
n1
n1
1
2n  1
2. How many terms of the series do we need to add in order to
find the sum if the error ≤.1?

 1
n 1
n 1
1
2n  1
Absolute and Conditional Convergence

n 1

1

an converges absolutely (is absolutely
A series 
n 1
convergent) if the corresponding series of absolute values,




an , converges. If
n 1

n 1
an

converges then
 1
n 1
n 1
an
converges.


 1
1 2
Ex. 
n
n 1
n 1


Ex.
 1
n 1
n 1
1
3
n



If


n 1

1

an

an
diverges but n 1
series is conditionally convergent.
n 1

converges, then the