~ ?yvD ?Dy
?
B?
f
Z 7?
a
)
# ?¨
~ ?yvD ?Dy
Baf
Z 7?
a)
°¤E
ü à)
E
??,;
!
)(
)1(
0
)(
n
xf
a
n
n
= p
,)(0lim)2(
)(
MxfR
n
n
n
≤=
∞→
)
).(xf ??5)

l ? uW
=
l
~ ?yvD ?Dy
è 1
3
.)( ￥
a)
Z 7?| xexf
x
=
,)(
)( xn
exf = ),2,1,0(.1)0(
)(
L== nf
n
LL +++++?
nx
x
n
xxe
!
1
!2
1
1
2
,0>?M
 ],[ MM?
xn
exf =)(
)(
M
e≤
),2,1,0( L=n
LL +++++=∴
nx
x
n
xxe
!
1
!2
1
1
2
?? M￥ ?i?,'¤
),(
!
1
!2
1
1
2
+∞?∞∈+++++= xx
n
xxe
nx
LL
~ ?yvD ?Dy
è 2
.sin)( ￥
a)
Z 7?| xxxf =
3 ),
2
sin()(
)(
π
+=
n
xxf
n
,
2
sin)0(
)(
π
=
n
f
n
,0)0(
)2(
=∴
n
f
,)1()0(
)12( nn
f?=
+
),2,1,0( L=n
=)(
)(
xf
n
O
)
2
sin(
π
+
n
x
1≤ ),( +∞?∞∈x
LL +
+
+?+?=∴
+
)!12(
)1(
!5
1
!3
1
sin
12
53
n
x
xxxx
n
n
),( +∞?∞∈x
~ ?yvD ?Dy
è 3
.)()1()( ￥
a)
Z 7?| xRxxf ∈+= α
α
3,)1)(1()1()(
)( nn
xnxf
α
++?α?αα= LQ
),1()1()0(
)(
+?α?αα= nf
n
L
),2,1,0( L=n
L
L
L +
+?α?αα
++
αα
+α+
n
x
n
n
xx
!
)1()1(
!2
)1(
1
2
n
n
n
a
a
1
lim
+
∞→
Q
1
lim
+
=
∞→
n
n
n
α
,1=
,1=∴R
~ ?yvD ?Dy
?
=,)1,1(?
L
L
L +
+?α?αα
++α+=
n
x
n
n
xxs
!
)1()1(
1)(
L
L
L +
+?α?αα
++?αα+α=
′
1
)!1(
)1()1(
)1()(
n
x
n
n
xxs
L
L
L +
+
++?+=
′
n
x
n
n
xxxsx
)!1(
)1()1(
)1()(
2
ααα
ααα
!
)1()1(
!
)()1(
)!1(
)1()1(
n
nmmm
n
nmm
n
nmm +
=

+
+ LLL
?¨
~ ?yvD ?Dy
)()1( xsx
′
+∴
L
L
L +
+?α?αα
++
αα
+α+α=
1
2
22
!
)1()1(
!2
)1(
n
x
n
n
xx
)(xsα=
,
1)(
)(
xxs
xs
+
=
′
∴
α
.1)0( =s O
Hs
,
1)(
)(
00
dx
x
dx
xs
xs
xx
∫∫
+
α
=
′
)1,1(?∈x
¤
),1ln()0(ln)(ln xsxs +α=?
~ ?yvD ?Dy
'
,)1ln()(ln
α
+= xxs
,)1()(
α
xxs +=∴
)1,1(?∈x
L
L
L +
+?α?αα
++
αα
+α+=
+∴
α
n
x
n
n
xx
x
!
)1()1(
!2
)1(
1
)1(
2
)1,1(?∈x
d=[
TZ 7
T
?i
.1 ￥ |′μ1)
l ??D α±=x
);1,1(1≤α
l ? uW1
];1,1(11?<α<?
l ? uW1
].1,1[1?>α
l ? uW1
~ ?yvD ?Dy
μ
H?,
2
1
,1 ±?=α
)1,1()1(1
1
1
32
+?++?+?=
+
LL
nn
xxxx
x
]1,1[
!)!2(
!)!32(
)1(
642
31
42
1
2
1
11
132
+
++

+
+=+
LL
nn
x
n
n
xxxx
]1,1(
!)!2(
!)!12(
)1(
642
531
42
31
2
1
1
1
1
32
+
++


+?=
+
LL
nn
x
n
n
xxx
x
¨e
~ ?yvD ?Dy
W¤E
? ·B?, ?¨ènZ 7
T,YV M
}D,
15
,?M?,?[ p?,?[s ?Z
E, pZ 7
T,
è ? )(sincos
′
= xx
LL +?+?+?=∴
)!2(
)1(
!4
1
!2
1
1cos
2
42
n
x
xxx
n
n
),( +∞?∞∈x
LLQ +
+
+?+?=
+
)!12(
)1(
!5
1
!3
1
sin
12
53
n
x
xxxx
n
n
~ ?yvD ?Dy
∫
+
=
x
x
dx
x
0
2
1
arctan
LL +
+
+?+?=
+
12
)1(
5
1
3
1
12
53
n
x
xxx
n
n
]1,1[?∈x
∫
+
=+
x
x
dx
x
0
1
)1ln(
LL +?+?+?=
n
x
xxx
n
n 132
)1(
3
1
2
1
]1,1(?∈x
~ ?yvD ?Dy
è 4
.4)(
45
￥
a)
Z 7?| xxxxf +=
3
2
1
2
)
4
1(2)(
x
xxf +=
+?+=
∑
∞+
=
+
n
n
n
x
n
nx
x
4!)!2(
!)!32(
)1(
42
1
12
2
12
1
4
1 ≤≤?
x
.
!)!2(4
!)!32(
)1(2
4
1
2
2
2
132 +
∞+
=
+
++=
∑
n
n
n
n
x
n
n
xx
44 ≤≤? x
]1,1[
!)!2(
!)!32(
)1(
642
31
42
1
2
1
11
32
+
++

+
+=+ LL
nn
x
n
n
xxxx
~ ?yvD ?Dy
è 5,2cossin)( ￥
a)
Z 7?| xxxxf =
3
xxxf 2cossin)( =
]sin3[sin
2
1
xx?=
LLQ +
+
+?+?=
+
)!12(
)1(
!5
1
!3
1
sin
12
53
n
x
xxxx
n
n
)!12(
)1(
2
1
)!12(
)3(
)1(
2
1
12
0
12
0
+

+
=
+∞+
=
+∞+
=
∑∑
n
x
n
x
n
n
n
n
n
n
+∞<<∞?
+
=
+
+∞+
=
∑
xx
n
n
n
n
n
.
)!12(
)13(
)1(
2
1
12
12
0
~ ?yvD ?Dy
è 6
.)1ln()(
2
￥
a)
Z 7?| xxxxf ++=
3
1
1
1
ln)(
3
≠
= x
x
x
xf
)1ln()1ln(
3
xx=
11)1()1ln(
1
1
≤<=+
∞+
=
∑
x
n
x
x
n
n
n
Q
n
x
n
x
n
n
n
n
n
n
)(
)1(
)(
)1(
1
1
3
1
1

=
∞+
=
∞+
=
∑∑
∑∑
∞+
=
∞+
=
=
1
3
1
.
n
n
n
n
n
x
n
x
11 <≤? x
~ ?yvD ?Dy
è 7
)Z 7?
ü à)
| 1
4
1
)( =
= x
x
x
xf
3
).1()1(
)(n
fx i p￥
a)
Z 7
)1(3
1
4
1

=
xx
Q
,
)
3
1
1(3
1
=
x
])
3
1
()
3
1
(
3
1
1[
3
1
2
LL +
++
+
+=
n
xxx
31 <?x
~ ?yvD ?Dy
x
x
x
x
=
∴
4
1
)1(
4
1
LL +
++
+
+?=
n
n
xxx
x
3
)1(
3
)1(
3
)1(
)1(
3
1
3
3
2
2
31 <?x
!
)1(
)(
n
f
n
?
^
.
3
!
)1(
)(
n
n
n
f =#
,
3
1
n
=
~ ?yvD ?Dy
=af
￥
a)
Z 7
T￥?¨
,
21
LLQ ++++=
n
aaaA
,
21 n
aaaA +++≈∴ L
.
21
L++=
++ nnn
aarμ
 ?ù5
1.ó?[
, pí
′i9ú;
2.óú, ??[
,
1r
YV9?[, ??ú[
,
aí
9
~ ?yvD ?Dy
è¨ZE,
1. ??[
^?p)
,5 V¨?￥
n[ ?3 % ;
2. ??
^?p)
,5bv??￥ò[,
P-?
1?1)
 
^ p￥)
,V7 p ,
è,10,
5?
P μ??V￥í
′9
e
3
,
!
1
!2
1
1
2
LLQ +++++=
nx
x
n
xxe
,1=x
7
,
!
1
!2
1
11
n
e ++++≈ L¤
~ ?yvD ?Dy
?,
L+
+
+
+
≈
)!2(
1
)!1(
1
nn
r
n
)
2
1
1(
)!1(
1
L+
+
+
+
=
nn
)
)1(
1
1
1
1(
)!1(
1
2
L+
+
+
+
+
+
≤
nnn
!
1
nn?
=
,10
5?
≤
n
rà
P
,10
!
1
5?
≤
nn
o1
,10!
5
≥?nn'
,10322560!88
5
>=?7
!8
1
!3
1
!2
1
11 +++++≈ Le
71828.2≈
~ ?yvD ?Dy
è
.
,9sin
!3
sin
0
3
i9μ
￥í
′9
?¨
x
xx?≈
3
20
sin9sin
0
π
=,)
20
(
6
1
20
3
ππ
≈
5
2
)
20
(
!5
1 π
≤r
5
)2.0(
120
1
<
300000
1
<,10
5?
<
000646.0157079.09sin
0
≈∴ 156433.0≈
μ??V,
5
10
~ ?yvD ?Dy
a9
?s
.,
,
ln
1
,
sin
,
2
4[9
?sf
V
U
ef
?
¨? è ?f
xx
x
e
x?
3E
?[s
?[s
Z 7?
a)
Z 7?
a)
?s￥í
′
?s￥í
′
$f
$f
~ ?yvD ?Dy
?
1[
3000
1
!77
1
<
,10
4?
<
| - ?[T1s￥í
′,¤
!55
1
!33
1
1
sin
1
0
+
≈
∫
dx
x
x
9461.0≈
è
.10,
sin
4
1
0
∫
ú ??￥í
′9
dx
x
x
LQ +?+?=
642
!7
1
!5
1
!3
1
1
sin
xxx
x
x
3
),( +∞?∞∈x
L+
+
=
∫
!77
1
!55
1
!33
1
1
sin
1
0
dx
x
x
l ?￥?p)
~ ?yvD ?Dy
a p
[)
￥
 ?¨)
￥?l p
(1)°¤E ;(2)·[E ;(3)?wE,
è
.
2
1
arctan
1
2
￥ p
∑
∞
=n
n
3
,
2
1
arctan
1
=s
8
1
arctan
2
1
arctan
2
+=s
8
1
2
1
1
8
1
2
1
arctan

+
=,
3
2
arctan=
~ ?yvD ?Dy
18
1
arctan
3
2
arctan +=
18
1
arctan
23
+= ss
,
4
3
arctan=
1arctan
1
arctan →
+
=∴
n
n
s
n
)(
4
∞→
π
= n
.
42
1
arctan
1
2
π
=
∑
∞
=n
n
#
,
1
arctan
1
k
k
s
k
=
L
!
2
2
1
arctan
1
arctan
kk
k
s
k
+
=,
1
arctan
+
=
k
k
~ ?yvD ?Dy
-:E /
a)
E

,lim
0
1
0
n
n
n
x
n
n
xaa
∑∑
∞
=
→
∞
=
=Q
,)(
0
n
n
n
xaxs
∑
∞
=
= p¤
).(lim
1
0
xsa
x
n
n
→
∞
=
=∴
∑
(?[sa?[ p? )
è,
2
12
1
￥ p
∑
∞
=
n
n
n
3
,
2
12
)(
22
1
∞
=
∑
=
n
n
n
x
n
xs
7 )2,2(?
~ ?yvD ?Dy
∑
∫
∞
=
′
=
1
0
22
)
2
12
()(
n
x
n
n
dxx
n
xs
∑
∞
=
′
=
1
12
)
2
(
n
n
n
x
))
2
(
1
(
1
2
′
=
∑
∞
=n
n
x
x
)
2
1
(
2
2
′
=
x
x
x
)
2
(
2
′
=
x
x
,
)2(
2
22
2
x
x
+
=
22
2
1
)2(
2
lim
x
x
x
+
=
→
)(lim
1
xs
x
→
,3=,3
2
12
1
=
∑
∞
=n
n
n
#
~ ?yvD ?Dy
è,
2!
1
2
￥ p
∑
∞
=n
n
n
n
3
,
!
)(
1
2
n
n
x
n
n
xs
∑
∞
=
=
7
),( +∞?∞
n
n
x
n
nnn
xs
∑
∞
=
+?
=
1
!
)1(
)(Q
n
n
n
n
x
n
x
n
nn
∑∑
∞
=
∞
=
+
=
11
)!1(
1
!
)1(
∑∑
∞
=
∞
=
+
′′
=
01
2
!
)
!
(
n
n
n
n
n
x
x
n
x
x
xx
xeex +
′′
= )1(
2
,)1( xxe
x
+=
∑
∞
=
∴
1
2
2!
n
n
n
n
)
2
1
(s=
2
1
)1
2
1
(
2
1
+= e
.
4
3
e=
~ ?yvD ?Dy
a
x ?
T
ˉ
[)
,
LL +++++++ )()()(
2211 nn
ivuivuivu
.),3,2,1(,1
Lè

Lf
? L=nvu
nn
?
∑
∞
=
=
1n
n
uu,
∑
∞
=
=
1n
n
vv,
5?)
∑
∞
=
+
1
)(
n
nn
ivu
l ?
 O 1 ivu+ 
~ ?yvD ?Dy
? LL +++++++
222
2
2
2
2
1
2
1 nn
vuvuvu
l ?
5
∑
∞
=1n
n
u
∑
∞
=1n
n
v '
l ?
?ˉ
[)
'
l ?
ˉ
[)
'
l ?￥à
Q
??'Z 7
T
,
!!2
1
2
LL +++++=
n
xx
xe
n
x
,
)!12(
)1(
!5!3
sin
12
1
53
LL +
+?+?=
n
xxx
xx
n
n
,
)!2(
)1(
!4!2
1cos
242
LL +?+?+?=
n
xxx
x
n
n
)( +∞<<?∞ x
)( +∞<<?∞ x
)( +∞<<?∞ x
~ ?yvD ?Dy
￥
a)
Z 7
T?
x
e
LL +++++=
nix
ix
n
ixixe )(
!
1
)(
!2
1
1
2
)
)!12(
)1(
!3
1
(
)
)!2(
)1(
!2
1
1(
12
3
2
2
LL
LL
+
+
++?+
+?++?=
+
n
x
xxi
n
x
x
n
n
n
n
.sincos xix+=
xcos
xsin
~ ?yvD ?Dy
xixe
ix
sincos +=Q
=
+
=
∴
i
ee
x
ee
x
ixix
ixix
2
sin
2
cos
xixe
ix
sincos?=
Q?
￡
U
 ??f
ˉM
·
f
-W￥
B?1"
x ?
T
)sin(cos yiyee
xiyx
+=
+
~ ?yvD ?Dy
?al2
1.f
Z 7?
ü à)
￥ZE ;
í
9
 p? V ?f
￥?s
±sZ?￥
a)
￥3E
p
[)
￥
x ?
T￥￡
ü
~ ?yvD ?Dy
± I5
?¨
a)
Z 7
T, pK
.
sin
arcsin
lim
3
0
x
xx
x
→
~ ?yvD ?Dy
± I53s
,
542
31
32
1
arcsin
53
L+?
+?+=
xx
xx
,
!5
33
!3
33
4
1
sin
5
5
3
3
3
+
= Lxxx
)1( ≤x
)( ∞<x
,
sin
arcsin
lim
3
0
x
xx
x
→
|


T} ?
~ ?yvD ?Dy
+?
+?+?
→
L
542
31
32
1
lim
53
0
xx
xx
x
+
L
5
5
3
3
!5
33
!3
33
4
1
xx
e
T =
)(
)(
6
1
lim
33
33
0
xox
xox
x
+
+?
=
→
.
6
1
=
~ ?yvD ?Dy
Ba|/
f
Z 7? x￥
a)

i pZ 7
T? ?￥ uW
a
x
a  a )1ln()1( xx ++ 
a xarcsin  a
3
)1(
1
x
x
+

=a|f
3
)( xxf = Z 7? )1(?x ￥
a)

i pZ 7
T? ?
￥ uW
?a|f
23
1
)(
2
++
=
xx
xf Z 7? )4( +x ￥
a)

1a|)
∑
∞
=
1
12
1
1
)!12(2
)1(
n
n
n
n
n
x
￥f
Z 7? )1(?x ￥
a)

5B
~ ?yvD ?Dy
5Bs?
Baa )(
!
)(ln
0
+∞<<?∞
∑
∞
=
xx
n
a
n
n
n

a )11(
)1(
)1(
1
1
1
≤<?
+
+
∑
∞
=
+
xx
nn
x
n
n
n

a )11()
2
(
)12()!(
)!2(2
1
12
2
≤≤?
+
+
∑
∞
=
+
x
x
nn
n
x
n
n

a )1,1(
1
12
∑
∞
=
n
n
xn 
=a +?+ )1(
2
3
1 x
∑
∞
=
+
++
0
2
2
)
2
1
(
2)2)(1(
3
)!(
)!2(
)1(
n
n
n
n
x
nnn
n
)20( ≤≤ x 
~ ?yvD ?Dy
?a )2,6()4)(
3
1
2
1
(
0
11
+?
∑
∞
=
++
n
n
nn
x 
1a
∑
∞
=
+
0
2
)1(
)!12(2
)1(
2
1
sin2
n
n
n
n
x
n
 ),()1(
)!12(2
)1(
2
1
cos
0
12
+∞?∞?
+
+
∑
∞
=
+
n
n
n
n
x
n

~ ?yvD ?Dy
Ba ?¨f
￥
a)
Z 7
T p/
ò
￥í
′
a 3ln  ú ?? 0001.0

a
o
2cos  ú ?? 0001.0

=a ?¨$f
￥
a)
Z 7
T p?s
∫
5.0
0
arctan
dx
x
x
 ú ?? 001.0
￥í
′
?a |f
xe
x
cos Z 7? ￥
a)
x 
5=
~ ?yvD ?Dy
5=s?
Baa a
=a
?a ),(
!4
cos2cos
0
2
+∞?∞?=
∑
∞
=n
n
x
n
xn
xe
π
π

 4
U
xi
xix
eexe
)
4
sin
4
(cos2
)1(
ReRecos
ππ
+
+
==