SECTION 8:
ALTERNATING SERIES
ALTERNATING SERIES:
A series in which the terms
alternate sign.


1

n
an
THE ALTERNATING
SERIES TEST
Series converges if:
1) Alternates
2) Terms decrease in
magnitude
3) Terms approach zero
EX 1: CONVERGE OR DIVERGE?
A)


n 0
 1
n 1
n!
B)


n 0
 3
n
EX 1: CONVERGE OR DIVERGE?
C)


n 0
cosn 
2
n 1
D)

 sin n 
n 1
ALTERNATING SERIES
ERROR BOUND:
If   1 an and
ak 1  ak for all k ,
then S k  ak 1  S n  S k  ak 1
n
EX 2:
USE S 5 TO GIVE BOUNDS
FOR THE VALUE OF:


n 0
 1
n
n!
EX 3:
Using the alternating series
bound, what degree
Maclaurin polynomial is
required to estimate cos(3)
with an error of no more
than 0.001?
• If  an converges,
then  an
is Absolutely Convergent.
•If  an diverges and  an
converges, then  an
is Conditionally Convergent.
EX 4: Absolutely Convergent,
Conditionally Convergent or
Divergent?
A)
B)
n
n

  1
1

n 0
n!

n 0
 
n 1
EX 4: Absolutely Convergent,
Conditionally Convergent or
Divergent?
C)


n 0
n n3
 1
n4
EX 5: FIND THE INTERVAL OF
CONVERGENCE:
  x  5

n
n 1 n  3
n
SECTION 8 WS