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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
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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
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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
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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
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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
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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. 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
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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
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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
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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 -
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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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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
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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
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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
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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!
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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
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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
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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
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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
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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
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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
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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
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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
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7/25/2019 CAL2 1 file.pdf
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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]
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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)
7/25/2019 CAL2 1 file.pdf
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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
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