Back and Forth

I'm All Shook Up...

The 3 R'sCompared to What?!

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ID, Please! - $200

ANSWER

1

2

3

n

n

¥

=

æö÷ç ÷ç ÷÷çè øå

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ANSWER

( )0

1.075n

n

¥

=å

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ANSWER

4 31

1

n n

¥

=å

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ANSWER

21 1n

n

n

¥

= +å

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ANSWER

1 1 1 11

2 2 3 3 4 4 5 5+ + + + +

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ANSWER

21

1

5 7n n

¥

= +å

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ANSWER

3 21 8 6 7n

n

n n

¥

= + -å

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ANSWER

31

1

2n n n

¥

= +å

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ANSWER

0

2

3 1

n

nn

¥

= +å

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ANSWER

( )2

1

ln

n

n

n

¥

=å

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ANSWER

1

1

2 1

n

n

n

n

¥

=

æ ö+ ÷ç ÷ç ÷÷çè ø+å

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ANSWER

31

2n

n n

¥

=å

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ANSWER

1

n

n

e¥

-

=å

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ANSWER

( )

2

ln

n

n

n

¥

=å

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ANSWER

( )1

!

1 3 5 2 1n

n

n

¥

=×× -å

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ANSWER

1

1 1

1n

n n

¥

=

æ ö÷ç - ÷ç ÷÷çè ø+å

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ANSWER

5 71

1

n n

¥

=å

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ANSWER

31 2n

n

n n

¥

= +å

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ANSWER

1

2 4 6 2

!n

n

n

¥

=

××å

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ANSWER

1

1

1 nn e

¥

-= +å

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ANSWER

( )

1

1n

nn

¥

=

-å

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ANSWER

( ) 1

1

1

2 1

n

n

n

n

+¥

=

-

-å

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ANSWER

( )2

1

1

1

n

n

n

n

¥

=

-

+å

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ANSWER

( )3

1

1n

n

n

n

¥

=

-å

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ANSWER

( )1

1cos

n

nn

p¥

=å

****Answers****

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DONE

A Convergent Geometric Series since r < 1.

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DONE

A Divergent Geometric Series since r > 1.

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DONE

A Divergent p – Series since p < 1.

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DONE

The limit of this series equals 1, therefore by the nth term test, the series DIVERGES.

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DONE

32

1 1

1 1

n nn n n

¥ ¥

= =

=å å

This series can be written as

which is just a convergent p – series, since p > 1

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DONE

21

1

n n

¥

=å

Using the Direct Comparison Test to the

convergent series

Therefore, this series converges also.

2 2

2 2

5 7

1 1

5 7

n n

n n

+ >

<+

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DONE

33 2

3 22

18 6 7lim lim 01 88 6 7n n

nnn n

n nn

®¥ ®¥

+ - = = >+ -

Use the Limit Comparison Test with 21

1

n n

¥

=å

Since converges, so does the original series.21

1

n n

¥

=å

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DONE

32

1

1

n n

¥

=å

Using the Direct Comparison Test to the

convergent p – series

Therefore, this series converges also.

3 3

3 3

2

1 1

2

n n n

n n n

+ >

<+

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DONE

1 1

2 2

3 3

n n

nn n

¥ ¥

= =

æ öæö ÷ç÷ç ÷ç=÷ç ÷ç÷÷ ÷ç ÷è ø ççè øå å

Using the Direct Comparison Test to the

convergent geometric series

Therefore, this series converges also.

3 1 3

1 1

3 1 3

2 2

3 1 3

n n

n n

n n

n n

+ >

<+

<+

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DONE

Using the Direct Comparison

Test, we first try

BUT this gives us a series that is less than a divergent series … Not helpful

Next, try using the Direct Comparison Test with

BUT this gives us a series that is more than a convergent series … Not helpful

So try using the Direct Comparison Test with something in between … say

2 2

ln

ln 1

n n

n n

nn n

<

< =

1

1

nn

¥

=

å

1

2

1

n n

¥

=

å

2 2

ln as long as 3

ln 1

1 n n

n

n n

>

>

>

1

32

1

n n

¥

=

å1

2

12

322 2

ln

ln 1

n n

n n

n n n

<

< =

Therefore, since the series is less than a convergent series, the original series is CONVERGENT ALSO!

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DONE

By the Root Test,

1 1 1lim lim 1

2 1 2 1 2

n

nn n

n n

n n®¥ ®¥

æ ö æ ö+ +÷ ÷ç ç= = <÷ ÷ç ç÷ ÷÷ ÷ç çè ø è ø+ +

this series converges.

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DONE

By the Ratio Test,

( )

( )

1

3 3

3

3

21 2

lim lim 2 12 1

n

nn n

n n

nn

+

®¥ ®¥

+= = >

+

this series diverges.

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DONE

this series converges because the integral converges.

By the Integral Test,

( )1

11

10x xe dx e e

e

¥¥- - -=- = - - =ò

NOTE: you could have used the geometric series test since

1 1

1n

n

n n

ee

¥ ¥-

= =

æö÷ç= ÷ç ÷÷çè øå å

The 3 R's - $800

DONE

this series diverges because the integral diverges.

By the Integral Test,

( )22

22 2 2

ln 1 1ln

2 2

xx

x x

xdx u du u x

x

®¥¥ ®¥ ¥

= =

= = =ò ò

The 3 R's - $1000

DONE

By the Ratio Test,

( )( )( )

( ) ( )( )

1 !1 3 5 2 1 1 1 1

lim lim 1! 22 1 11 3 5 2 1

n n

nn n

n nn

®¥ ®¥

+×× + - +

= = <+ -

×× -

this series converges.

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DONE

By the Telescoping Series Test, this series converges.

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DONE

A convergent p – series, since p > 1

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DONE

Using the Limit Comparison Test to the

divergent p – series

Therefore, since the p – series diverges, so does the original series.

12

3

32

3

12

lim lim 1 0

2n n

n nnn

nn n

®¥ ®¥

+= = >

+

12

1

1

n n

¥

=å

I'm All Shook Up... - $800

DONE

By the Ratio Test,

( )( )( ) ( )

( )

2 4 6 2 11 ! 2 1

lim lim 2 12 4 6 2 1

!n n

nn n

n nn

®¥ ®¥

×× ++ +

= = >×× +

this series diverges.

I'm All Shook Up... - $1000

DONE

1 1lim lim lim 1

11 11

n

n nn n nn

e

e ee

-®¥ ®¥ ®¥= = =

+ ++

By the nth term Test, since

this series diverges

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DONE

Alternating Series Test

(condition 1):

(condition 2):

1lim 0n n®¥

=

1

1 1

1

n n

n n

+ >

<+

Since both conditions of the alternating series are met, the series converges.

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DONE

Alternating Series Test

(condition 1):

Since the first condition of the alternating series test fails, then what we just did is the equivalent of the nth term test.

Therefore, this series diverges.

1lim

2 1 2n

n

n®¥=

-

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DONE

Alternating Series Test

(condition 1):

(condition 2): To show it is decreasing, find the 1st derivative

The 1st derivate is 0 for x = 1, but negative everywhere else. Therefore, both conditions are met and the series converges.

2lim 0

1n

n

n®¥=

+

( )

( )( )( ) ( )( )

( ) ( )

2

2 2

2 22 2

1

1 1 2 1'

1 1

xf x

x

x x x xf x

x x

=+

+ - - += =

+ +

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DONE

Alternating Series Test

(condition 1): 1

6

3lim limn n

nn

n®¥ ®¥= =¥

Since the first condition of the alternating series test fails, then what we just did is the equivalent of the nth term test.

Therefore, this series diverges.

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DONE

First, recognize this as an Alternating Series … since

(condition 1):

(condition 2):

Since both conditions of the alternating series are met, the series converges.

1lim 0n n®¥

=

1

1 1

1

n n

n n

+ >

<+

( ) ( ) ( ) ( )1

cos 1 1 1 1 1 1n

np¥

=

= - + + - + + - + +å

CONTINUE

CONTINUE

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• To change the questions and answers, just type over the problems…Use the “replace” feature to change the categories easily

• The daily doubles were originally set to category #1 for $800 and category #5 for $400

• To change the daily doubles you must– 1. Change the hyperlink for the links on the main board to go to the

appropriate question, therefore bypassing the daily double slide … (right click on category #1 for $800 and chose hyperlink, then edit hyperlink)

– 2. There is a rectangle drawn around each slide that is used to link the slide question with the answer (or answer back to the main board). To change this link, position your cursor at the edge of the slide to select the rectangle and edit the hyperlink by right clicking the rectangle. You need to edit the hyperlink on each Daily Double slide.