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