CAL2 ลำดับและอนุกรม 1 file.pdf

7/25/2019 CAL2 1 file.pdf

1/41

2

8 165

8

1.(Sequence)

1.1

8.1.1(Sequence){1, 2, 3, , n}{1, 2, 3, }

f ( )f n nf1f (1), (2), (3), ..., ( ), ...f f f f n

2

f

1 2 3

, , , ..., , ...n

a a a a

1 2

(1), (2), ..., ( )n

a f a f a f n

3f{ }n

a n

a n{ }n

a

8.1.1 n

1. 1, 4, 9, 16 2. 3, 5, 7, 9, 3.1 1 1 1

, , , , ...3 4 5 6

1. 1, 4, 9, 16, 2 2 3 21 , 2 , 3 , 4 , ... n 2n

2. 3, 5, 7, 9, 2(1) 1, 2(2) 1, 2(3) 1, 2(4) 1, ... n 2 1n

3. 1 1 1 1, , , , ...3 4 5 6

1 1 1 1, , , , ...1 2 2 2 3 2 4 2

n 12n


2/41

2

8 166

8.1.2 4

1. 22 n 2.1

1

n

n

3. 1

1

n n

n

1. 22 n

1 = 22 1 = 3 2 = 22 2 = 63 = 22 3 = 11 4 = 22 4 = 184 22 n 3, 6, 11, 18

2. 11

n

n

1 = 1 11 1

= 0 2 = 1 22 1

=1

3

3 =1 3

3 1

=1

2 4 =1 4

4 1

=3

5

4 11

n

n

1 1 30, , ,

3 2 5

3. 11

n n

n

1 = 1 111 1

=

1

2 2 = 2 21

2 1

=

2

3

3 = 3 313 1

=3

4 4 = 4 41

4 1

=4

5

4 11

n n

n

1 2 3 4, , ,

2 3 4 5

1.2

8.1.2(Arithmetic Sequence)(common different)

1 2 3 1, , , ..., , , ...n na a a a a d 1n nd a a

n

1n na a d

1

( 1)n

a a n d


3/41

2

8 167

8.1.3 50 19, 12, 5, 2, 9, ... 9 2 7

1

19a

n

a =1

( 1)a n d

50

a =1

49a d = 19 49(7) = 324

503248.1.4 15, 12, 9, 6, 12 15 3

1

15a

n

a =1

( 1)a n d

na = 15 ( 1)( 3)n = 18 3n = 3(6 )n

18 3n 3 6 n

1.3 8.1.3(Geometric Sequence)

(common ratio)

1 2 3 1, , , ..., , , ...

n na a a a a r 1n

n

ar

a

n1n n

a a r

11

n

na a r

8.1.5 7 1 2 4, , , ...3 9 27

r= 23

1

1

3a

na = 11 na r

7

a = 61

a r =

61 2

3 3

=64

2187

8.1.6 27, 9, 3, 1, ...

r= 13

1 27a

na = 11 na r

na =1

1( 27)

3

n

=1

813

n

1813

n


4/41

8

1.4

n

8.1.7

1

n

a

8.1.8

1, na

a

n

3

2 , 2,2

a

n 2 , 2, 8 ,

11

n

4, 3, , 4

3

2 1

2 n

3, 32 , 4,

n

5

, ..., ,4

n

128 , ..., n

( , )nn a

11 , ...

n

1

( , )nn a

2 12 , ...n

n

n

limn

n

a

n

n

n

1

lim

nn

a

2

168

na

na


5/41

8

2 12 n

x

8.1.9 na

1,

na

8.1.4

8.1.5

L

8.1.7

cos n

a

n

1 , 2,1

8.1.7

na lim

nn

a L

n N

na

na

na na

l

n

cos( )n

3, 1 , 4

-1

1

8.

cos n

na a

M

M

m ( ) li

xf n

8.1.9

,1 , ,..., n

li

n

1.9

(conver

(di

0

(di0

( )f x

( , )nn a

os( ) , ...n

na

1

ent sequen

0

(d

erge to

verge to

x

11

n

n

1

n

e)

vergent se

)

N

) N

LN

uence)

limn

na

n M

lim nn a

n M

f

22 n

2

169

)x

1


6/41

2

8 170

8.1.1 na nb 1. lim limn n

n nk a k a

2. lim lim limn n n nn n n

a b a b

3. lim lim limn n n nn n n

a b a b

4.lim

limlim

nn n

nn n

n

aa

b b

lim 0

nn

b

8.1.10 3 9 12, 2, , , ...2 4 5

3 9 12, 2, , , ...2 4 5

31

n

na

n

3 3 3lim lim 311 1 0

1n n

n

n

n

3

8.1.2 2, , , ..., , ...na ar ar ar

1.

1r

0

2. 1r ()3. 1r 4. 1r

8.1.11

1.21

5

n

n

2.

ln n

n

3. 1 2 3 4, , , , ...2 3 4 5

4. 1 1 1 11, , , , , ...2 4 8 16

5.5 8 11 14

, , , , ...2 3 4 5

6.5 2 7 4 9 8 11 16

, , , , ...1 2 3 4

7. 1

2 nn 8.3 2

3

n

n

1. 21

5n

na

n

2 2

2

11

1lim lim lim

1 55n

n n n

n nan

n n


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2

8 171

2. lnn

na

n lnlim lim

nn n

na

n

1

ln 1lim lim lim 0

1n n n

n n

n n 0

3. 1 2 3 4, , , , ...2 3 4 5

1

n

na

n

1 1lim lim lim 111 1 0

1n

n n n

na

n

n

1

4. 1 1 1 11, , , , , ...2 4 8 16

1

1

2

n

na

1 1

12 2r

0

5. 5 8 11 14, , , , ...2 3 4 5

3 11

n

na

n

1

33 1 3 0

lim lim lim 311 1 0

1n

n n n

n nan

n

3

6. 5 2 7 4 9 8 11 16, , , , ...1 2 3 4

2(4 6 )( 1)n

n

n na

n

2(4 6 )

lim lim( 1) limnn

n n n

n na

n

lim( 1)n

n

7. 1

2 nn

a n 1

lim lim 2 nn

n na n

0

1

11 lim ln(2 ) lim01

lim 2 1

n n

nnnn

n n e e e

1

8. 3 23

n

na

n

23

3 2lim lim lim

3 31

nn n n

nn n

an

n


8/41

2

8 172

8.11. 1

7 11 17, ,a a a 3

3

2. 95008003.

1 2, ,u a u a b n2

1 22n n nu u u

2

2 1n nu au

4. 18 32

7298

5. n1,024520

6. 51, 3 5273

242565(1)7. 2, 6, 18, 1,0008. 1 2 3 4, , ,a a a a 4 1a

2 36a a

3 412a a 5

9. a, b, c27 , 3, 2a b c a b c

10.x,y,z121, 4, 11xyz

11. na n 11 1

2 5n na a na

12.

1)2 1

n

n

2)

3

2 1

n

n

3)

2

3 1

n

n

4)

2

3 1n

n

5)2

2

1

1

n

n

6)

2

2

1( 1)

1

nn

n

7)

2

3

1( 1)

1

nn

n

8) 1 ( 1)n

9)1

2

1)n

n

10)

n

n

e

11)

1ln

n

12)

2n

n

13) sinn

14) cos

2n

15)

4

n

n

16)2

n

17) cos2

n

18) 2 nn e 19) 2 3n n n 20) 2n n

21)2

3 ( 1)n

n

22)2

2

ln( )n

n

23) 12

tan2 1

n

n

24)1 2 3

... n

n n n n


9/41

2

8 173

2.(Series)

8.2.1 1 2 3, , ,..., na a a a n 1 2 3 ... na a a a

1

n

ii

a nS

1 2 3, , ,..., ,...

na a a a

1 2 3... ...

na a a a

1

i

i

a

2.1 (capital sigma)

1.1

n

i

c cn

c

2.1 1

n n

i i

i i

ca c a

c

3. 1 1 1

n n n

i i i i

i i i

a b a b

4.1

( 1)1 2 3 ...

2

n

i

n ni n

5. 2 2 2 2 2

1

( 1)(2 1)1 2 3 ...

6

n

i

n ni n

6.

23 3 3 3 3

1

( 1)1 2 3 ...

2

n

i

n ni n

8.2.1 1 3 3 5 5 7 ... 21 23

1 3 3 5 5 7 ... 21 23 =11

1

(2 1)(2 1)i

i i

=11

2

1

(4 1)i

i

=11 11

2

1 1

4 1i i

i

=(11 1)(22 1)

4 1(11)6

= 2013

8.2.2

2 2 2 21 2 2 3 3 4 ... 19 20

2 2 2 21 2 2 3 3 4 ... 19 20 =19

2

1

( 1)i

i i

=19

3 2

1

( 2 )i

i i i

=19 19 19

3 2

1 1 1

2i i i

i i i

=

219(19 1) (2 1)(2 2 1) 19(19 1)

22 6 2

= 36,100 + 4,940 + 190

= 41,230


10/41

2

8 174

2.2

8.2.2 1 2 3, , ,...,

na a a a d

12n nn

S a a

12 12

n

nS a n d

8.2.3 1 6 11 16 ... 101 1 6 11 16 ... 101

1 1a , 5d 101na

1( 1)

na a n d

101 = 1 + (n 1)(5)n= 21

21

n 12

n n

nS a a

1 6 11 16 ... 101 = 21 1 1012

= 1071

2.3

8.2.3 1 2 3, , ,..., na a a a 1r

1 , 11

nn

a a rS r

r

1(1 ) , 11

n

n

a rS r

r

8.2.4 2 4 8 16 ... 2048 2 4 8 16 ... 2048

1 2a , 2r 2048na 1

1

n

na a r

2048 =1

(2)(2)n

2048 = 2n

n= 11

11

n 1 ; 11

nn

a a rS r

r

2 4 8 16 ... 2048 = 2 (2048)(2)1 2

= 4094

8.2 -


11/41

2

8 175

3.3.1

8.3.1

1 2 3

, , ,...a a a

1 2 3

...n n

S a a a a (partial sum) n

1 2 3, , , ...S S S

8.3.1 nS lim nn

S S

(convergent series)S S nS lim n

nS

(divergent series)

1.:

1 ( 1)na a n d

10a d

2.: 11 nna a r - 1r

- 1r 11

aS

r

1. 1

n

n

a

lim 0n

na

lim 0

nn

a

1

n

n

a

2. lim 0nn a

1

n

n

a

3. 1

n

n

a

1

n

n

b

1

n n

n

a b

4. 1

n

n

a

1

n

n

b

1

n n

n

a b

5. 1

n

n

a

1

n

n

b

1

n n

n

a b


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2

8 176

8.3.1

1.1

1

n n

2.

1 1n

n

n

3.

1

1

2

3

n

n

4.

1

1

53 4

n

n

5.

2 2 2 2...1 3 9 27 6. 1 5 25 125 ...

1. 1

1

n n

1nS

n 1lim lim 0

nn n

Sn

0

2. 1 1n

n

n

1n

nS

n

1lim lim lim 1

111

nn n n

nS

n

n

1

3. 1

1

23

n

n

1 1a

2 21

3 3r

1 1 1 32 11

13 3

aS

r

3

4.

1

1

53

4

n

n

13a

5 51

4 4r

5. 2 2 2 2 ...1 3 9 27

1

1

12

3

n

n

1

2a 1 1 13 3

r

1 2 2 31 211

3 3

aSr

3

6. 1 5 25 125 ... 11

5n

n

1

1a 5 5 1r


13/41

2

8 177

3.2 (P-series)

8.3.2 (P-series)1

1 1 1 1...

1 2 3p p p pn n

p

1p (harmonic series)

1 1 1 11 ... ...2 3 4 n

12

p

1 1 1 11 ... ...

2 3 4 n 1p

2 2 2 2

1 1 1 11 ... ...

2 3 4 n 2p

2 2 2 21 2 3 4 ... ...n 2p

8.3.2 1

1p

n n

1p 1

1p

n n

1p 1

1p

n n

8.3.2

1. 1 1 11 ...2 3 4

2.2 2 2

1 1 11 ...

2 3 4

3.1 1 1

1 ...2 2 3 3 4 4

1. 1 1 11 ...2 3 4

1 12

p

2. 2 2 21 1 11 ...2 3 4 2 1p

3. 1 1 11 ...2 2 3 3 4 4

3 12

p


14/41

2

8 178

3.3 (Comparison Test)

8.3.3 1

n

n

a

1

n

n

b

0na 0nb

1, 2, 3, ...n

1. 1

n

n

b

n na b n

1

n

n

a

2. 1

n

n

b

n na b n

1

n

n

a

8.3.3

1.1

1

1n n

2.

2

21

sin

1n

n

n

3.

5

2

31

1

n

n

n

4.

1 1 1 1...

1 3 2 5 3 7 4 9

5.1 lnn

n

n

6. 1 1 11 ...

2! 3! 4!

1. 1 1

1n n

2, 3, ...n

1

1

n n

1

2p

1

1

1n n

2. 1 sin 1n 1, 2, 3, ...n

20 sin 1n 2 1n 2

2 2

sin 10

1 1

n

n n

(1)

2 2

1 1

1n n

1, 2, 3, ...n (2)

(1)(2)2

2 2 2

sin 1 1

1 1

n

n n n

1, 2, 3, ...n

2

1

1

n n

2p

2

21

sin

1n

n

n


15/41

2

8 179

3. 55

32

3 3 3 3

1 1 1 1 1n n

n n n nn n

1, 2, 3, ...n

1

1

n n

1

2p

5

2

31

1

n

n

n

4. 1 1 1 1 ...1 3 2 5 3 7 4 9

1

1

(2 1)n n n

2 2

1 1 1 1

(2 1) 2 2n n n n n

1, 2, 3, ...n

2

1

1

n n

2p

1 1 1 1 ...1 3 2 5 3 7 4 9

5. ln n n 1, 2, 3, ...n

1 1ln n n

1

ln

n n

n n n

1

1

n n

1p

1 lnn

n

n

6. 1 1 11 ...2! 3! 4!

1

1

!n n

1

1 1 1 1

! 1 2 3 ... 1 2 2 ... 2 2nn n

1, 2, 3, ...n

1

1

1

2nn

1

2r

1 1 1

1 ...2! 3! 4!


16/41

2

8 180

3.4 (Limit comparison Test)

8.3.4 1

n

n

a

1

n

n

b

0na 0nb

1, 2, 3, ...n

1. lim 0nn

n

aL

b

2. lim 0nn

n

a

b

1

n

n

b

1

n

n

a

3. lim nn

n

a

b

1

n

n

b

1

n

n

a

8.3.4

1.3

51

2 5

4 1n

n

n

2.

2

31

4 5

2n

n n

n n

3.2

1 1

n

nn

e

e

4.

1

3

1

n

nn e

5.2

1

sin2 1n n

6. 2

1

1

lnn n n

1. 3

5

2 5

4 1n

na

n

2

1nb

n

3

5 25

5

2

2 52 5 14 1lim lim lim 0

1 4 1 2

n

n n nn

na n nn

b n

n

2

1

1

n n

2 1p

3

51

2 5

4 1n

n

n

2. 2

3

4 5

2

n

n na

n n

1nb

n

2

3 23

3

4 5

4 52lim lim lim 1 01 2

n

n n nn

n n

a n n nn n

b n n

n

1

1

n n

2

31

4 5

2n

n n

n n


17/41

2

8 181

3. 21

n

n n

ea

e

1

n nb

e

22

2

1lim lim lim 1 0

1 1

n

nnn

nn n nn

n

e

a ee

b ee

1

1n

n e

11r

e

2

1 1

n

nn

e

e

4. 31

n

n na

e

3

n

n nb

e

3

1lim lim lim 1 03 1

n

nn

nn nn n n

n

n

a ee

b e

e

1

3n

nn e

3 1r

e

1

3

1

n

nn e

5. 2

sin2 1

na

n

22 1n

bn

2

2

sin 2 1lim lim 1 0

2 1

n

n nn

a n

b

n

2 2

1 1

1

2 1n nn n

21

1

n n

( 2 1p )

2

1 2 1n n

2

1

sin2 1n n

6. 2

1

lnn

an n

2

1n

bn

2

2

11lnlim lim lim 0

1 ln

n

n n nn

a n n

b n

n

2

1

1

n n

2 1p

21

1

lnn n n


18/41

2

8 182

3.5 (Integral Test)

8.3.5 1

n

n

a

0na nf

,a ( ) 0f x x a ( ) nf n a n

1.1

n

n

a

( )

a

f x dx

2.1

n

n

a

( )

a

f x dx

8.3.5

1. 21

1

n n

2. 11

n n

3.2

21

1 1sech

n n n

1. 2

1n

an

2

1( )f x

x

2

11 1

1 1 1( ) lim ( ) lim lim 1 1

b

b b b

b

f x dx dxx x b

1

( )

f x dx 2

1

1

n n

2. 1

na n 1

( )f x x

1

1 1

1( ) lim ( ) lim 2 lim 2 2

b

b b b

b

f x dx dx x bx

1

( )

f x dx

1

1

n n

3. 22

1 1sech

na

n n 2

2

1 1( ) sechf x

x x

2 221 1 1

1 1 1 1( ) lim ( sech ) lim (sech ) ( )

b b

b b

f x dx dx dx x x x

1

1 1lim tanh lim tanh tanh1 tanh1

b

b bx b

1

( )

f x dx 22

1

1 1sech

n n n


19/41

2

8 183

3.6 (Ratio Test)

8.3.6 1

n

n

a

0na 1lim n

nn

a

a

1. 1 1

n

n

a

2. 1 1

n

n

a

3. 1

8.3.6

1.1

( 1) 2

!

n

n

n

n

2.

1

( 3)!

3 !n

n

n

n

3.2 3

3 5 71 ...

5 5 5 4.

1 1 3 1 3 5 1 3 5 7...

5 5 6 5 6 7 5 6 7 8

5.3 3 5 3 5 7 3 5 7 9

...2 2 7 2 7 12 2 7 12 17

1. ( 1) 2!

n

n

na

n

( 1)

1

( 1 1) 2 ( 2) 2 2

( 1)! ( 1) !

n n

n

n na

n n n

1 ( 2) 2 2n

n

n

a n

a

( 1) !n n

!n( 1) 2

nn

2

2 4

2 1

n

n n

2

1

2

2

2 42 4 0 0

lim lim lim 0 12 12 1 1 0 0

1

n

n n nn

a n n n

a n n

n n

1

( 1) 2

!

n

n

n

n

2. ( 3)!3 !

n n

na

n

1 ( 1)

( 1 3)! ( 4)! ( 4)( 3)!

3 ( 1)! 3 3 ( 1)! 3 3 ( 1) !n n n n

n n n na

n n n n

1

( 4) ( 3)!n

n

n na

a

3 3n ( 1) !n n

3 !n n

( 3)!n

4 4

3( 1) 3 3

n n

n n

14

14 1 0 1

lim lim lim 133 3 3 0 3

3

n

n n nn

a n n

a n

n

1

( 3)!

3 !nn

n

n


20/41

2

8 184

3. 1 1

2 1 2 1

5 5 5n n n

n na

1 ( 1) 1 1 1

2( 1) 1 2 2 1 2 1

5 5 5n n n n

n n na

1 2 15

n

nn

a n

a

5n

15 1 2 1

2 1 5 2 1

n

n n

112

1 2 1 1 1 2 0 1lim lim lim 1

15 2 1 5 5 2 0 52

n

n n nn

a n n

a n

n

2 3

3 5 71 ...

5 5 5

4. 1 3 5 ... (2 1)5 6 7 ... ( 4)

n

na

n

1

1 3 5 ... (2 1)(2 1)

5 6 7 ... ( 4)( 5)n

n na

n n

1(2 1)

n

n

na

a

(2 1)

( 4)

n

n

( 4)

( 5)

n

n

(2 1)n

2 1

5

n

n

11

22 1 2 0

lim lim lim 2 155 1 0

1

n

n n nn

a n n

a n

n

1 1 3 1 3 5 1 3 5 7 ...5 5 6 5 6 7 5 6 7 8

5. 3 5 7 ... (2 1)2 7 12 ... (5 3)

n

na

n

1

3 5 7 ... (2 1)(2 3)

2 7 12 ... (5 3)(5 2)n

n na

n n

1(2 1)

n

n

naa

(2 3)

(5 3)

n

n

(5 3)

(5 3)

n

n

(2 1)n

2 35 3

nn

13

22 3 2 0 2

lim lim lim 135 3 5 0 5

5

n

n n nn

a n n

a n

n

3 3 5 3 5 7 3 5 7 9 ...2 2 7 2 7 12 2 7 12 17


21/41

2

8 185

3.7 n(nth-Root Test)

8.3.7 1

n

n

a

0na limn n

na R

1. 1R 1

n

n

a

2. 1R 1

n

n

a

3. 1R

8.3.7

1.1

2 n

n n

2.1

6 3

3 5

n

n

n

n

3.

1

1

ln( 2) n

n n

1. 2 n

na

n

2 2n

n nna

n n

2lim lim 0 1n nn n

R an

1

2 n

n n

2. 6 33 5

n

n

na

n

6 3 6 33 5 3 5

n

n nn

n na

n n

6 3 6lim lim 2 13 5 3

nn

n n

nR a

n

1

6 3

3 5

n

n

n

n

3.

1

ln( 2)n n

an

1 1

ln( 2)ln( 2)

nnn n

ann

1lim lim 0 1ln( 2)

nn

n nR a

n

1

1

ln( 2) n

n n


22/41

2

8 186

3.8 (Alternating Series Test)

8.3.3 (Alternating Series)

1

1 2 3 4

1

( 1) ...n nn

a a a a a

0, 1, 2,3,...ia i

8.3.8 11 2 3 4

1

( 1) ...n nn

a a a a a

0na

21.

1n na a n

2. lim 0n

na

8.3.8

1.1 2 3 4

...3 7 11 15

2. 12

1

2( 1)

4

n

n

n

n

1. 4 1

n

na

n

(1) 11

4 3 4 1n n

n na a

n n

n

(2) 1 1 1lim lim lim 014 1 4 0 4

4n

n n n

nan

n

1 2 3 4 ...3 7 11 15

2. 2

2

4n

na

n

(1)1 2 2

3 2

2 5 4n n

n na a

n n n

n

(2)2

2

2

1 2

2 0 0lim lim lim 044 1 0

1n

n n n

n n nan

n

12

1

2( 1)

4

n

n

n

n


23/41

2

8 187

3.9 (Absolutely Convergent & Conditionally Convergent)

8.3.4 1

n

n

a

1

n

n

a

1

n

n

a

1

n

n

a

1

n

n

a

8.3.9

1. 13

1

1( 1)

(2 1)

n

n n

2.

1

21

5( 1)n

n n n

1. 3

1

(2 1)na

n

(1) 1 3 3

1 1

(2 3) (2 1)n na a

n n

(2)3

1lim lim 0

(2 1)n

n na

n

3

1 1

1

(2 1)n

n n

an

31 1

1

(2 1)n

n n

an

3 3 3

1 1 1

(2 3) (2 ) 8n n n

n

31

1

(2 )n n

3p

1

n

n

a

13

1 1

1( 1)

(2 3)

n

n

n n

an

2. 2

5na

n n

(1) 1

2 2

5 5

3 2n na a

n n n n

(2)2

5lim lim 0nn n

an n

2

1 1

5n

n n

an n

2

1 1

5n

n n

an n

2 2

5 5 5

nn n n

n

1

5

n n

1p

1

n

n

a

1

21 1

5

( 1)n

n

n na

n n


24/41


25/41

2

8 189

8.31.

1 2 3, ,S S S

4S

1)1

1

5

5

n

n

2) 11

5

2

n

n

3)1

1

2

4

n

n

4)1 1n

n

n

5)

1

1

( 1)( 2)n n n

6)

1

1

2

3

n

n

2.

1)1

1

5nn

2)

1

1

3

4

n

n

3)2

1

2

3

n

n

4)

1

1

32

4

n

n

5)1

5

3

n

n

e

6) 21

1n

n e

7) 11

1

7( 1)

6

n

nn

8)1

1

3

2

n

n

9)2

11

4

7

n

nn

10) 1

1

3 8n n

n

11)2

1

3 11

3( 1)

2

nn

nn

12)1 3 4n

n n

n n

13)1

1

( 2)( 3)n n n

14) 2

1

1

9 3 2n n n

15) 1

1

1 1

2 2n nn

16)1 2n

n

n

17)

2

1 2 1n

n

n

18)

2

21

1

2 1n

n n

n

19)1

sin( )n

n

20)

1

cos( )n

n

21)

1

tan( )n

n

22)1

ln ( )n

n

23)

1

1ln

n n

24)1

ln1n

en

n

25)1

n

n

e

26)

1

1

n

n

e

27)

1

n

n

e

n

28)1

1

4n n

29)

1

1

2 10n n

30)

1

5

4n n

31)1

110n n

32) 1 12 ( 1)n n n

33) 1 103 ( 1)n n n

3.

1)1

n

n

ne

2) 2

1 1n

n

n

3)

1

1

lnn n n

4)

1

1

2 1n n

5)

3

1

1

2 1n n

6)

1

1sin

n n n

7)

4

51 9n

n

n

8)

2

3 31 10n

n

n

9)

1

ln

n

n

n

10) 2 31

( 1)n

n

n


26/41

2

8 190

4. ()

1)1

1

4nn n

2) 2

1

1

1n n

3)

1

1

( 2)n n n

4) 3

1

1

n

n

n

5)2

1

1n

n e

6) 3 2

1

1

n n n

7)

1

1 5

4

n

nn

8)2 1n

n

n

9)3

1

1

1n

n

n

10) 1

3 sin

3nn

n

11)

51 1n

n

n

12)

1

2

1 3

n

nn

13)1

1 2

1 3

n

nn

14)

2

41

1

1n

n

n

15) 1

1

1n n

16)

2

1

sin

n

n

n n

5. ()

1)1

1

1n n

2)

2

41

1

1n

n

n

3)

3 2

4 21

4 3

1n

n n

n n

4)1

1

2nn

n

n

5)

2

21

2 7

3 (3 4 )nn

n n

n n

6) 21

5

2 5n

n

n

6. nn

1)1

3 1

2nn

n

2)

1

(1 )n n

n

e

3)

11

5

(3 )

n

nn n

4)

1

7

!

n

n n

5)1

100

!

n

n n

6)

2

1 5n

n

n

7)

1

!n

n

n

e

8) 50

1

n

n

n e

9)3

1

2n

n n e

10)

1

2

3

n

n

n

11) 21

3

4

n

n n

12)

2

1

lnn n n

13)3

1

!

n

n

n

14)

1

4

7 1

n

n n

15) 21

4n

n n

16)

2

1

( !)

(2 )!n

n

n

17)1

3 2

2 1

n

n

n

n

18)

1

lnn

n

n

e

19)1 5

n

n

n

20)1

( 1)!

4! !4n

n

n

n

21)1 2 1 2 3 1 2 3 4

1 ...1 3 1 3 5 1 3 5 7

22)

2! 3! 4! 5!...

1 1 4 1 4 7 1 4 7 10

7.

1)1

2( 1)

3

n

n n

2)

1

1

( 1)1

n

n

n

n

3) 1

1( 1)n

nn e

4) 1 21

1( 1)n

n n

5)1

( 1)

3

n

n n

6) 1

1

( 1)5 1

n

n

n

n

7)

1

1

1( 1)

ln

n

n n n

8) 21

( 1)1

n

n

n

n

9)

2

21 ( 1) 1

n

n

n

n

10) 1 ( 1) 2n

nn

n

11) 1 ( 1) lnn

n

n

n

12)1

1

ln

( 1)

n

n

n

n


27/41

2

8 191

13) 1

1

1( 1)

3 1

n

n

n

n

14) 31 4cos

n

n

n

15)

1

( 1) sinn

n n

16)1

( 1) cosn

n n

17)1

32

( 1)

ln

n

n n

18) 2

1

( 1)n

n

n

e

n

19)1

3

5

n

n

20)2

1

( 1)nn

n

n

e

8.

1) 1

1

1( 1)n

n n n

2) 1 31

1( 1)n

n n

3)1

2( 1)

1

n

n

n

n

4) 12

1

( 1)1

n

n

n

n

5) 31 1n

n

n

6)

1

2

3 4n

n

n

7)1

2( 1)

3 4

n

n

n

n

8)

1

1

1( 1)

(2 )!

n

n n

9) 31

( 3)n

n n

10)1

1( 1)(2 1)!

n

n n

11) 21 3

1( 1)

( 1)

n

nn

12) 31 2

1( 1)

( 1)

n

nn

13)1

( 1)n n

n

e

14)1

( 1)n n

n

e

15) 11

1( 1)

!

n

n n n

16)1

( 1)ln

n

n

n

n

17)1

ln( 1)n

n

n

n

18) 31

sin2

n

n

n

19)2

1

1cos

6

n

n

n

20)1

cos

n

n

n


28/41

2

8 192

4.(Power Series)

8.4.1 0x 2

0 1 20

... ...n n

n nn

a x a a x a x a x

x h

2

0 1 2

0

( ) ( ) ( ) ... ( ) ...n nn n

n

a x h a a x h a x h a x h

h(Center)0 1 2, , ,..., ,...na a a a

0x

0

n

n

nx

na n 0n

1

n

n

x

n

1na

n 1n

2

0 (2 )!

n

n

x

n

1

(2 )!na

n 0n

x h

0

( 2)n

n

x

1na 0n 2h

1

( 2)

!

n

n

x

n

1

na n 1n 2h

8.4.2 (Interval of Converge) ( , )a b , [ , ]a b , ( , ]a b [ , )a b

8.4.3 Rx

x a R a R R(Radius of Converge)x(R )x a 0


29/41

2

8 193

8.4.1 20 1 2

0

... ...n nn n

n

a x a a x a x a x

x c ( 0c ) x c

8.4.2 20 1 2

0

... ...n nn nn

a x a a x a x a x

x d x d

1nx a R a R x a R

2

8.4.1 2 3 4

...2 3 4

x x xx

n

n

xa

n

1

11 1

n n

n

x x xa

n n

1lim limn

n

n nn

a x xa

1 nn

n x

lim

11

n

x x

n

1x 1 1x 1x 1x

1x 1

( 1) 1 1 11 ...

2 3 4

n

n n

1x 1

1 1 1 11 ...

2 3 4

n

n n

( 1p ) 1 1x


30/41

2

8 194

8.4.2 3 5 7

...3 5 7

x x xx

2 1 1 2

2 1 2 1

n n

n

x x xa

n n

2 1 2

12 1 2 1

n n

nx x xa

n n

2

1lim limn

n

n nn

a x x

a

1 2

2 1

2 1 nn

n x x

2 2 22 1lim2 1n

nx x x

n

2 1x 1 1x 2 1x 1x

1x 2 1

1

( 1) 1 1 11 ...

2 1 3 5 7

n

n n

1x 2 1

1

1 1 1 11 ...2 1 3 5 7

n

n n

( 1p ) 1 1x

8.4.3 1

( 1) 3 ( 2)n n n

n

x

n

( 1) 3 ( 2)n n n

n

xa

n

1 1 1

1

( 1) 3 ( 2)

1

n n n

n

xa

n

1( 1) ( 1) 1n n

1 1

1 3 ( 2)lim lim 3lim ( 2) 3 21 3 ( 2) 1

n n

n

n nn n nn

a x n nx x

a n x n

3 2 1x 5 73 3

x

3 2 1x 5 7,3 3

x x

53x 2

1 1

1( 1) 3

( 1) 1 1 13 1 ...2 3 4

n

n n

n

n nn n

73

x 1 1

1( 1) 3

( 1) (1) 1 1 131 ...

2 3 4

n

n n

n n

n nn n

5 73 3

x


31/41

2

8 195

8.4

1.

0 4

n

n

x

n

2.2

0 2

n

n

n

x n

3.1

1

( 1)n

n

n

x

n

4. 22 1

n

n

n x

n

5.3

2

lnn

n

x n

n

6.

0 1

n

n

x

n

7.0

( 1)

!

n n

n

x

n

8. 21

5n n

n

x

n

9.1 ( 1)

n

n

x

n n

10.

2 1

0

( 1)(2 1)!

nn

n

x

n

11.0

3

!

n n

n

x

n

12. 2

0 1

n

n

x

n

13.0

n

n

nx

14.

0 !

n

n

x

n

15.1

( 1)

2

n n

nn

x

n

16. 20

3

( 1)

n n

n

x

n

17.2 ln

n

n

x

n

18.

0

! n

n

n x

19.1 3

n

nn

xn

20.

1

0

( 1)2 1

n n

n

xn

21.1

1

( 1)( 1)

nn

n

x

n

22.

0

3( 5)

4

n

n

n

x

23.2 1

21

( 1)( 1)

4

nn

n

x

n

24.2

0

( 1)

(2 1)!

n n

n

x

n

25.0

1( 4)

10

n

nn

nx

26.

2

0

3( 2)

1

nn

n

xn

27.

2

30

( 4)2

n

nn

nx

28. 1

ln( )nn

n

nx ee

29.1

( 6)2

n

n

n

nx

30. 31

(2 1)n

n

x

n


32/41

2

8 196

5.(Taylor and Maclaurin Series)

Brook Taylor (1685-1731) Colin Maclaurin (1698-1746)

8.5.1 f 0x

( ) ( )2

0

(0) (0) (0) (0)

(0) ... ...! 1! 2! !

n nn n

n

f f f f

x f x x xn n

(Maclaurin Series)

8.5.2 fx a

( ) ( )2

0

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ... ( ) ...

! 1! 2! !

n nn n

n

f a f a f a f ax a f a x a x a x a

n n

(Taylor Series)

8.5.1 2( ) xf x e

( )

2(0) (0) (0)(0) ... ...1! 2! !

nnf f f

f x x xn

0( ) (0) 1xf x e f e 0( ) (0) 1

xf x e f e

0( ) (0) 1xf x e f e

( ) ( ) 0( ) (0) 1n x nf x e f e

2 3

0

1 ...2! 3! ! !

n n

n

x x x xx

n n

!

n

n

xa

n

1

1( 1)!

n

n

xa

n

1

1 ! 1lim lim lim 0 1

( 1)! 1

n

n

nn n nn

a x nx

a n x n

x


33/41

2

8 197

8.5.2 ( ) sinf x x

( )

2(0) (0) (0)(0) ... ...1! 2! !

nnf f f

f x x xn

( ) sin (0) sin 0 0f x x f

( ) cos (0) cos 0 1f x f

( ) sin (0) sin 0 0f x x f

( ) cos (0) cos 0 1f x x f

(2 ) (2 )( ) ( 1) sin (0) ( 1) sin 0 0

n n n nf x x f

(2 1) (2 1)( ) ( 1) cos (0) ( 1) cos0 ( 1)n n n n nf x x f

3 5 7 2 1

( 1)... ...

3! 5! 7! (2 1)!

n nx x x x

xn

8.5.3 2( )x

f x e 2a

( )

2( ) ( ) ( )( ) ( ) ( ) ... ( ) ...1! 2! !

nnf a f a f a

f a x a x a x an

2

2 2( ) (2)x

f x e f e e 2

2 21 1 1

( ) (2)2 2 2

x

f x e f e e

2

2 21 1 1

( ) (2)

4 4 4

x

f x e f e e

( ) ( )21 1

( ) (2)2 2

n nx

n nf x e f e

21 1 1( 2) ( 2) ... ( 2) ...2 4 2! 2 !

n

ne e x e x e x

n

8.5.4 ( ) lnf x x 2a

( )

2( ) ( ) ( )( ) ( ) ( ) ... ( ) ...1! 2! !

nnf a f a f a

f a x a x a x an

( ) ln (2) ln 2f x x f 1 1

( ) (2)2

f x fx

2

1 1( ) (2)

4f x f

x

1 1

( ) ( )( 1) ( 1)( ) (2)2

n nn n

n nf x f

x

1

21 1 ( 1)

ln 2 ( 2) ( 2) ... ( 2) ...2 4 2! 2 !

nn

nx x xn


34/41

2

8 198

8.51.

1) ( )f x xe 2) ( )f x sin3x

3) ( )f x cos( )x 4) ( )f x cosh2

x x

e ex

5) ( )f x 2

1 cos2

xx 6) ( )f x 2xe

7) ( )f x ln(3 2 )x 8) ( )f x sinxe x

9) ( )f x 21 cos 2

sin2

xx

10) ( )f x 2x

11) ( )f x 3 8 x 12) ( )f x cos ln(1 )x x

13) ( )f x 1

cosh

1

x

x

14) ( )f x 2sec sinx x x

2. x a

1) ( ) ln , 1f x x a 2)1

( ) , 2f x ax

3) ( ) sin ,4

f x x a

4) ( ) , 4f x x a

5) ( ) , 1f x e a 6) 1( ) tan , 1f x x a

7) 2( ) 4 2 1, 1f x x x a 8) ( ) ln , 2f x x a 9) ( ) ln ,f x x a e 10) ( ) tan ,f x x a


35/41

2

8 199

81. n

1)1 2 3

ln ln ln ... ln ...2 3 4 1

n

n

2) 1

1

1 1 1 11 ... ( 1) ...

2 4 8 2

n

n

3)1

2 2 2 22 ... ...

3 9 27 3n 4)

5 5 5 5... ...

1 2 2 3 3 4 ( 1)n n

2.

1)1 1 1

1 ... ...3 9 3

n

2)

2 31 1 1

... ...7 7 7

n

3)1

1

1 1 1 ( 1)1 ... ...

2 4 8 2

n

n

4)

5 6 73 3 3 3

... ...4 4 4 4

n

5)

1

1 2

n

n

6) 11

7 3

n

n

3.

1)3

1

1

n n

2)

1

1

n n

3) 1

1n

n

4)

2

3

1n

n

5)

4

3

1n

n

6) 4

1

1

n n

7)3 5

1

1n n

8)

1

1n n

9)1 1 1

1 ...2 2 3 3 4 4

10)1 1 1

1 ...4 2 9 3 16 4

4.

1)1

ln

n

n

n

2) 2

1

1

1n n

3)2

1

1

1n n n

4)

1

1

2 1n n

5)1

1

lnn n n

6) 2

1 1n

n

n

5.

1)1

2

2 1n n

2)

1

2

3 1nn

3)1

1

3 5nn

4)

1

2 1

3 2

n

nn n

5)1

9

1n n

6)

2

1

sin

2n

n

n


36/41

2

8 200

6.

1)3

1

2 3

n

n

n

2)3 2

4 31

2 3 4

5 2 1n

n n

n n

3)1

1

( 1)( 2)n n n

4)2

1

3 2

2 5n

n

n

5)1

3 4

2nn

n

n

6) 11

5n

n

n

n

7.

1)1

2

3

n

n

n

2)1 (2 2)!n

n

n

3)3

1 !n

n

n

4)

1 2n

n

n

5)1

n

n

ne

6)

3

21

23

n

nn

7)2

2

1

( 1)

3

n

nn

n

8) 21 2

n

nn

n

9)1

!(2 1)!

(2 3)!n

n n

n

10) 1

1 4 7 ... (3 2)

3 5 7 ... (2 1)n

n

n

8. n

1)1 5

n

n

n

2)2

1

1

3

n

n

n

n

3)1

1

ln

n

n n

4)1 5

nn

n

5)1 (ln )

nn

n

n

6) 2

1

( !)

( )

n

nn

n

n

9.

1) 12

1

1( 1)n

n n

2)1

1( 1)

ln

n

n n n

3) 1

1

1( 1)n

n

nn

4)2

1

21

( 1)5 2

n

n

nn

5) 1

1

1( 1)

2 1

n

n n

6)

1

1

1( 1)

3 1

n

n

n

n

10.

1) 1

1

( 1) (0.1)n n

n

2) 11

1( 1)n

n n

3) 13

1

( 1)1

n

n

n

n

4) 1

!

2nn

n


37/41

2

8 201

5)1

1( 1)

3

n

n n

6) 21

sin( 1)n

n

n

n

7) 1

1

3( 1)

5

n

n

n

n

8)1

31

1( 1)

ln

n

n n

9) 12

1

1( 1)n

n

nn

10)1

1

( 2)5

n

nn n

11) 2

1

2

3

n

n

n

12) 11

( 1) ( 10)n n

n

13)1

21

tan( 1)

1

n

n

n

n

14)

1

1

1( 1)

ln

n

n n n

15)1

1 1

2n n n

16) 11

(0.1)( 1)

nn

n n

11.1)

0

( 5)n

n

x

2)0

( 2)n

n

x

n

3)0

(2 )n

n

x

4)

0

( 1) ( 2)n n

n

x

n

5)0

( 1)n

n

x

n

6)0

3

!

n n

n

x

n

12.1) ( ) xf x e 2) ( ) cosf x x

3) ( ) (1 )mf x x 4) ( ) sinhf x x 5) ( ) coshf x x 6) ( ) tanf x x

7) sin( ) xf x e 8) ( ) ln(cos )f x x

13.1) 3( )f x x 2x 2) 4 2( ) 3f x x x 1x 3) ( ) lnf x x 1x 4) ( ) xf x e 3x


38/41

2

8 202

8.11. 19 2. 33 3. 2 2( 1)n b 4. 75. 5120 6. 341205 7. 1458 8. 48

9. 13 10. 60-420 11.2

5

12. 1) : 12

2) : 32

3) 4) : 0 5) : 1 6)

7) : 0 8) 9) : 0 10) : 0 11) 12) : 013) : 0 14) : 1 15) : 0 16) 17) 18) : 0

19) : 32

20) : 0 21) : 0 22) : 0 23) :4

24)

8.3

1. 1)1 2 3 4

24 124 6244, , ,

5 25 125

S S S S 2)1 2 3 4

5 15 35 75, , ,

4 8 16 32

S S S S

3)1 2 3 4

1 3 7 15, , ,

4 4 4 4S S S S 4)

1 2 3 4

1 7 23 163, , ,

2 6 12 60S S S S

5)1 2 3 4

1 1 3 1, , ,

6 4 10 3S S S S 6)

1 2 3 4

1 7 131, , ,

3 9 27S S S S

2. 1) : 14

2) : 47

3) : 89

4) : 8 5) : 53

e

e 6) :

2

1

1e

7) : 6 8) 9) : 4483

10) 11) 12) : 14

13) : 13

14) : 16

15) 16) 17) 18) : 0

19) 20) : 0 21) 22) 23) 24) 25) 26) 27) 28) 29) 30)

31) 32) : 12

33) :103

3. 1) 2) 3) 4) 5) 6) 7) 8) 9) 10)

4. 1) 1

1

4nn

2) 2

1

1

n n

3) 2

1

1

n n

4) 212

n n

5) 11

nn e

6) 3

2

1

n n

7)

1

5

4

n

n

8) 2

1

n n

9) 2

1

1

n n

10) 1

1

2

3nn

11) 3/21

1

n n

12)

1

2

3

n

n

13) 1

22

3

n

n

14) 21

2

n n

15)

1

1

2n n

16) 1

1

n n n


39/41

2

8 203

5. 1) 1

1

n n

2) 2

1

1

n n

3)

1

1

n n

4) 1

1

2nn

5)

1

1

3nn

6)

1

1

n n

6. 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12)13) 14) 15) 16) 17) 18) 19) 20) 21) 22)

7. 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20)

8. 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20)

8.4 1. 0 1 [ 1,1) 2. 0 2 ( 2, 2)3. 0 1 ( 1,1] 4. 0 1 [ 1,1)5. 0 1 [ 1,1] 6. 0 1 [ 1,1)

7. 0 R 8. 01

5

1 1,

5 5

9. 0 1 [ 1,1] 10. 0 R11. 0 R 12. 0 1 [ 1,1]13. 0 1 ( 1,1) 14. 0 R

15. 0 2 ( 2, 2] 16. 01

3

1 1

,

3 3

17. 0 1 [ 1,1) 18. 0 0 {0}19. 0 3 [ 3,3) 20. 0 1 ( 1,1]

21. -1 1 ( 2,0] 22. -54

3

19 11,

3 3

23. -1 1 [ 2,0] 24. 1 R

25. 4 10 ( 6,14) 26. 21

9

17 19,

9 9

27. -4 8 ( 12, 4) 28. e e (0, 2 )e

29. -6 0 {-6} 30.1

2

1

2

[0,1]


40/41

2

8 204

8.5

1. 1)2 3

1 ...2! 3!

x xx 2)

3 5 7(3 ) (3 ) (3 )3 ...

3! 5! 7!

x x xx 3)

2 4 6

1 ...2! 4! 6!

x x x

4)2 4 6

1 ...2! 4! 6!

x x x 5)

4 6 8 10

...4! 6! 8! 10!

x x x x 6) 2 3 4

4 21 2 2 ...

3 3x x x x

7) 2 32 2 8

ln3 ...3 9 81x x x 8)

3 52 ...

3 30

x xx x 9)

2 3 4 5 6 7 82 2 2 2...

2! 4! 6! 8!

x x x x

10)0

(ln 2)

!

n n

n

x

n

11) 3 1

0

1

( 1) 32

nn

nn

x

n

12)2 3 53

...2 6 40

x x xx

13)2 3 43 3 37

1 ...2 3 24

x x xx 14)

3 5612 ...

6 120

x xx

2.1) 2 3

1 1( 1) ( 1) ( 1) ...

2 3x x x 2) 2 3

1 1 1 1( 2) ( 2) ( 2) ...

2 4 8 16x x x

3) 22 2 2

( ) ( ) ...2 2 4 4 4

x x

4) 2 31 1 1

2 ( 4) ( 4) ( 4) ...4 64 512

x x x

5) 2 3( 1) ( 1) ( 1) ...2! 3!

e ee e x x x 6) 2 3

1 1 1( 1) ( 1) ( 1) ...

4 2 4 12x x x

7) 243 26( 3) 4( 3) ...x x 8)1

1

( 1) ( 2)ln 2

2

n n

nn

x

n

9)2 3

2 3

( ) ( )1 ...

2 3

x e x e x e

e e e

10) 3

1( ) ( ) ...

3x x

8

1. 1) ln( 1)nS n , 2)2 1 2

1 , lim3 2 3

n

n nn

S S

3)1

3 1 , lim 33

n

n nn

S S

4)

55 , lim 5

1n n

nS S

n

2. 1) : 32

2) : 142

3) : 43

4) : 4 5) 6) : 7

2

3. 1) 2) 3) 4) 5) 6) 7) 8) 9) 10)

4. 1) 2) 3) 4) 5) 6) 5. 1) 2) 3) 4) 5) 6) 6. 1) 2) 3) 4) 5) 6) 7. 1) 2) 3) 4) 5) 6)

7) 8) 9) 10) 8. 1) 2) 3) 4) 5) 6)


41/41

2

9. 1) 2) 3) 4) 5) 6) 10. 1) 2) 3)

4) 5) 6) 7)

8)

9)

10) 11) 12) 13) 14) 15) 16)

11. 1) 6 4x 2) 1 3x 3)1 1

2 2x

4) 3 1x 5) 0 2x 6) x

12. 1)2 3 4

1 ...2! 3! 4!

x x xx 2)

2 4 6

1 ...2! 4! 6!

x x x

3) 2 3( 1) ( 1)( 2)1 ...2! 3!

m m m m mmx x x 4)3 5 7

...3! 5! 7!

x x xx

5)2 4 6

1 ...2! 4! 6!

x x x 6)

3 52...

3 15

x xx

7)2

1 ...2

xx 8)

2 4 6

...2 12 45

x x x

13. 1)2 38 12( 2) 6( 2) ( 2)x x x 2) 2 31 6( 1) 7( 1) 4( 1) ...x x x

3)2 3( 1) ( 1)

( 1) ...2 3

x xx

4)

2 33 ( 3) ( 3)

1 ( 3) ...2! 3!

x xe x

CAL2 ลำดับและอนุกรม 1 file.pdf

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