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SERIES TESTS

a
Question in
the exam
n 1
2) Integral Test

lim an  0
n 


1
3) Comparison Test
Is the series convergent or divergent?
Special Series:
Series Tests
1)Divergence Test
n
 ar
1)Geometric Series
convg : r  1
n 1

f ( x)dx
2)Harmonic Series

n 1
bn  an
4) Limit Compar Test c  lim
n 
3)Telescoping Series
an
bn
divg
1
n

 (b b
n 1
n
n 1
)
4)Alter Harmonic

1
n
5) Ratio Test
5)p-series
6)Root Test
6)Alternating p-series
7)Alter Series Test
n 1
n 1
p
convg : p  1
ALTERNATING
SERIES TEST
ALTERNATING SERIES
series with positive terms
Series Tests
1 1 1 1 1
1     
2 3 4 5 6
1)Divergence Test
2) Integral Test
lim an  0
n 


1
3) Comparison Test
series with some positive
and some negative terms
5) Ratio Test
6)Root Test
alternating series
n 1

1 1 1 1 1
( 1)
1       
n
2 3 4 5 6
n 1
bn  an
4) Limit Compar Test c  lim
n 
All
1 1 1 1 1 1
1      
2 3 4 5 6 7
f ( x)dx
7)Alter Series Test
an
bn
ALTERNATING SERIES
1 1 1 1 1
1     
2 3 4 5 6


n 1
( 1)
n 1
alternating series
n

( 1)
n 1
 1 n  odd

 1 n  even
a
n 1
n
n-th term of the series
an  (1)n  1un
un
are positive
ALTERNATING SERIES
alternating series


n 1
n 1
n 1
a

(

1
)
un
 n 
(1) n 1
1 1 1 1 1
 1     

n
2 3 4 5 6
n 1

alternating harmonic series
un 
1
n

1 n
1 1 1 1
( )      

2
2 4 8 6
n 1
(1) n1
1
1
1
1

1






p
p
p
p
p
n
2
3
4
5
n 1
alternating geomtric series
un 

1 n
2

alternating p-series
un 
1
np
ALTERNATING SERIES
THEOREM: (THE ALTERNATING SERIES TEST)
1) un  0
2 ) un1  un
un  0
3) lim
n

 (1)
n 1
alternating
n 1
un

 (1)
decreasing
n 1
lim = 0
Remark:
The convergence tests that we have
looked at so far apply only to series
with positive terms. In this section and
the next we learn how to deal with
series whose terms are not
necessarily positive. Of particular
importance are alternating series,
whose terms alternate in sign.
n 1
un
convg
Example:
Determine whether the series
converges or diverges.
(1) n1

n
n 1

ALTERNATING SERIES
THEOREM: (THE ALTERNATING SERIES TEST)
1) un  0
alternating
2 ) un1  un
decreasing
un  0
3) lim
n

 (1)
n 1

Example:
Determine whether the series
converges or diverges.

2
n
n 1
(

1
)

3
n
1
n 1
un
 (1)
n 1
lim = 0
n 1
n 1
un
convg
ALTERNATING SERIES
THEOREM: (THE ALTERNATING SERIES TEST)
1) un  0
alternating
2 ) un1  un
decreasing
un  0
3) lim
n

 (1)
n 1

un
 (1)
n 1
lim = 0
n 1
n 1
un
convg
Example:
Determine whether the series
converges or diverges.

3n
( 1)

4n  1
n 1
n
Alternating Series, Absolute and Conditional Convergence
the series of absolute values

 an  1 
n 1

 an  1 
n 1
1 1 1 1 1
    
2 3 4 5 6


n 1
1
1
1
1
1





2 2 32 4 2 5 2 6 2

a
n 1
(1) n 1
1 1 1
1
1

1







2
n
4
9
16
25
36
n 1

n
n 1
1
1
1
1
1





2
2
2
2
2
2
3
4
5
6
1
1 1 1
1
1

1







2
n
4
9
16
25
36
n 1
Example:

a
 1
1 1 1 1 1
    
2 3 4 5 6

DEF:
The series
an  1 
n
is called
Absolutely convergent (AC)
If the series of absolute values is convergent
a
n 1
(1) n1

2
n
n 1


Also we may say that
Test the series for absolute
convergence.
n
converges absolutely
Alternating Series, Absolute and Conditional Convergence
DEF:

a
n 1

n
Is called
Absolutely convergent
IF
a
n 1
convergent
converges absolutely
Example:
Test the series for absolute
convergence.

sin( n)

2
n
n 1
n
Alternating Series, Absolute and Conditional Convergence
DEF:

The series
a
n 1
n
is called
Conditionally convergent (CC)
If it is convergent but the series of absolute values is divergent
DEF:
Example:

a
n 1
n
Is called conditionally convergent
if it is convergent but not absolutely convergent.
REM:

a
n 1
convg
a
n 1
(1) n1

n
n 1


n
Test the series for absolute
convergence.
n
divg
Alternating Series, Absolute and Conditional Convergence

THM:
a
n 1
n
Absolutely
convergent

THM:
a
n 1

a
n 1
n
convergent
n
convg

convg
n
a
n 1
Example:
Determine whether the series
converges or diverges.
The signs change irregularly
cosn 

2
n
n 1

Alternating Series, Absolute and Conditional Convergence

a
n 1

 an
n 1

a
n 1
n
conditionally
convergent
Absolutely
convergent

a
n 1
n
convergent
n
divergent

a
n 1

n
a
n 1
n
SERIES TESTS

a
Questions
in the exam
n 1
Is the series convergent or divergent?
n
Special Series:

 ar
1)Geometric Series
n 1
convg : r  1
n 1

2)Harmonic Series

n 1

 (b b
3)Telescoping Series
n 1


4)Alter Harmonic
n 1

5)p-series
1
n
n 1
p
divg
1
n
( 1) n
n
n
n 1
)
CC
convg : p  1
6)Alternating p-series
(1) n

np
n 1

 AC : p  1
CC : p  1

ALTERNATING SERIES
series with positive terms
Series Tests
1 1 1 1 1
1     
2 3 4 5 6
1)Divergence Test
2) Integral Test
lim an  0
n 


1
3) Comparison Test
series with some positive
and some negative terms
5) Ratio Test
6)Root Test
alternating series
n 1

1 1 1 1 1
( 1)
1       
n
2 3 4 5 6
n 1
bn  an
4) Limit Compar Test c  lim
n 
All
1 1 1 1 1 1
1      
2 3 4 5 6 7
f ( x)dx
7)Alter Series Test
an
bn
THE RATIO
TEST
THE RATIO AND ROOT TESTS
Ratio Test

 an
be an infinite series
an 1
n  a
n
L  lim
n 1
AC


divg

the test is inconclusi ve

L 1
L 1
L 1
the Ratio Test is inconclusive; that is, no conclusion
can be drawn about the convergence or divergence
Example:
Test the series for
convergence.

n3

n
3
n 1
THE RATIO AND ROOT TESTS
TERM-132
Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS
TERM-082
THE ROOT
TEST
THE RATIO AND ROOT TESTS
Root Test

a
n 1
be an infinite series
n
AC


divg

the test is inconclusi ve

Example:
Test the series for
convergence.
 2n  3 



n 1  3n  2 

n
L 1
L 1
L 1
L  lim
n 
n
an
Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS
TERM-082
Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS
TERM-132
SERIES TESTS

a
Question in
the exam
n 1

lim an  0
1)Divergence Test
n 

2) Integral Test

1
3) Comparison Test
1)Geometric Series
f ( x)dx
bn  an
n 
n 1
convg : r  1
n 1

a
c  lim n
n  b
n
a
L  lim n 1
n  a
n
L  lim
 ar
2)Harmonic Series

n
an
7)Alter Series Test alt , dec, lim 0

 (b b
3)Telescoping Series
n 1


4)Alter Harmonic
n 1

5)p-series
1
n
n 1
p
divg
1
n
n 1
4) Limit Compar Test
6)Root Test
Is the series convergent or divergent?
Special Series:
Series Tests
5) Ratio Test
n
( 1) n
n
n
n 1
)
CC
convg : p  1
6)Alternating p-series
(1) n

np
n 1

 AC : p  1
CC : p  1

Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS
TERM-082
THE RATIO AND ROOT TESTS
TERM-101
Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS
TERM-091