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1
1p
n n
Lesson 4 – P-Series
1 2 3 #'Term s
1 1 1 1
1 2 3p p p pn
General Form of P-Series is:
1p P Series Converges
1p P Series Diverges
11
1
n n
Lesson 4 P-Series
1 2 3 4 5 6 7 8 #'Term s1 1 1 1 1 1 1 11 2 3 4 5 6 7 8 ...
A Harmonic Series is a P-Series with P=1
1
1
n
akan
1 1 1 1 1 1 1 11 2 3 4 5 6 7 8 ...
1 1 1 1 1 1 1 11 2 4 4 8 8 8 8 ...
1 1 1 11 2 2 2 ...
Sooooo, the Harmonic Series DIVERGES
Lesson 4 P-Series
The other way to deal with P-Series is with The Integral Test
.i f is positiveConditions for the integral test
11
,
( )nn
then
a and f x dx
. , ' 0iii f is decreasing that is f x .ii f is continuous
Either both converge
1 ( ) :nfor x and a f n
or both diverge
Lesson 4 P-Series
1 1 1...
1 2 3
Ex 1 Let’s use The Integral Test to to show that the Harmonic Series (p=1) DIVERGES
1
1
n n
1
( )Let f xx
Does f(x) meet the conditions?
. ?i Is f positive
. ?iii Is f decreasing
. ?ii Is f continuous
1, ( ) 0if x f x YES
YES
YES
Is ' 0?f x
2
1' ? 0f x
x
Lesson 4 P-Series
Sooooo, the Harmonic Series DIVERGES again!
Soooo, now let’s apply The Integral Test
1
1dxx
1ln( )x
ln ln 1
Lesson 4 P-Series
1 1 1...
1 4 9
Ex 2 Let’s use The Integral Test to test the convergence of the P-Series with P=2
21
1
n n
2
1( )Let f x
x Does f(x) meet the
conditions?
. ?i Is f positive
. ?iii Is f decreasing
. ?ii Is f continuous
1, ( ) 0if x f x YES
2
11, ( )if x f x YES
x
3
2'f x
x 0 YES
Lesson 4 P-Series
Sooooo, the P-Series with P=2 Converges
Soooo, now let’s apply The Integral Test
21
1dx
x
1
1
x
1 1( )
1
1
Lesson 4 P-Series
1 1 1...
1 2 3
Ex 3 Let’s use The Integral Test to test the convergence of the P-Series with P=1/2
1
1
n n
1
( )Let f xx
Does f(x) meet the conditions?
. ?i Is f positive
. ?iii Is f decreasing
. ?ii Is f continuous
1, ( ) 0if x f x YES 1
1, ( )if x f x YESx
' 0f x
3
1'
2f x
x 0 YES
Lesson 4 P-Series
Sooooo, the P-Series with P=1/2 DIVERGES
Soooo, now let’s apply The Integral Test
1
1dxx
12 x
2( 1)
1
2
1x dx
Lesson 4 P-Series
1 1 1...
1 8 27
Ex 4 Let’s use The Integral Test to test the convergence of the P-Series with P=3
31
1
n n
3
1( )Let f x
x Does f(x) meet the
conditions?
. ?i Is f positive
. ?iii Is f decreasing
. ?ii Is f continuous
1, ( ) 0if x f x YES
3
11, ( )if x f x YES
x
4
3'f x
x 0 YES
Lesson 4 P-Series
Sooooo, the P-Series with P=3 CONVERGES
1
2
Soooo, now let’s apply The Integral Test
31
1dx
x
21
1
2x
1 1 1
2 1
3
1x dx
Lesson 4 P-Series
1 2 3...
2 5 10
Ex 5 Let’s use The Integral Test to test the convergence of the Series:
21 1n
n
n
2( )
1
xLet f x
x
Does f(x) meet the conditions?
. ?i Is f positive
. ?iii Is f decreasing
. ?ii Is f continuous
1, ( ) 0if x f x YES
21, ( )
1
xif x f x YES
x
2
22
1 1 2'
1
x x xf x
x
2
22
1
1
x
x
0 YES
Lesson 4 P-Series
Sooooo, the Series DIVERGES
Soooo, now let’s apply The Integral Test
21 1
xdx
x
2
1ln( )
2u
2 1Let u x 2du x dx
2
dux dx
2
1 1
2duu
Lesson 4 P-Series
1 1 1...
2 5 10
Ex 6 Let’s use The Integral Test to test the convergence of the Series:
21
1
1n n
2
1( )
1Let f x
x
Does f(x) meet the conditions?
. ?i Is f positive
. ?iii Is f decreasing
. ?ii Is f continuous
1, ( ) 0if x f x YES
2
11, ( )
1if x f x YES
x
2
22
1 0 1 2'
1
x xf x
x
22
2
1
x
x
0 YES
Lesson 4 P-Series
Sooooo, the Series CONVERGES
4
Soooo, now let’s apply The Integral Test
21
1
1dx
x
1
1tan ( )x
1 1tan ( ) tan (1) 2 4