复旦大学：数学分析（习题全角、教学教案）：ex10-1

? Ec f ?[) ?  5 10. 1 f ?[) ?￥Bá l ?? 1. ) ?/ f ?? ·?uW ￥Bá l ??b ò S n (x) = , (i) x nx? e ∈ )1,0(
 (ii) x∈
),1( +∞ ó S n (x) = x , x nx? e ∈ ),0( +∞
? S n (x) = sin n x , (i) x∈ ),( +∞?∞
 (ii) x∈ ],[ AA? ( )
0>A ? S n (x) = arctan nx, (i) x∈ )1,0(
 (ii) x∈
),1( +∞ ? S n (x) = 2 2 1 n x + , x∈ ),( +∞?∞
× S n (x) = nx(1

x) n , x∈ ]1,0[
? S n (x) = n x ln n x , (i) x∈ )1,0(
 (ii) x∈ )
),1( +∞ ù S n (x) = n n x x +1 , (i) x∈ )1,0(
 (ii) x∈
),1( +∞ ú S n (x) = (sin x) n , x∈ ],0[ π
? S n (x) = (sin x) n 1 , (i) x∈[0, ]π
 (ii) x∈ ],[
0>δ

δπδ ? ü S n (x) = n n x ? ? ? ? ? ? +1 (i) x∈ ),0( +∞
 (ii) x∈ ],0( A ( )
0>A Y S n (x) = ? ? ? ? ? ? ? ? ?+ x n xn 1 , (i) x∈ ),0( +∞ , (ii) [ ) 0,, >+∞∈ δδx b 3
1
(i)
 0)( =xS )()(sup),( )1,0( xSxSSSd n x n ?= ∈ 1= a /1 0
∞→n

 ?[ { }() n Sx dBá l ?b (0,1) (ii)
 0)( =xS )()(sup),( ),1( xSxSSSd n x n ?= +∞∈ n e ? = )(0 ∞→→ n
 ?[ { }() n Sx Bá l ?b (1, )+∞
2

 0)( =xS )()(sup),( ),0( xSxSSSd n x n ?= +∞∈ ne 1 = )(0 ∞→→ n
 1 ?[ { }() n Sx Bá l ?b (0, )+∞
3
(i)
 0)( =xS )()(sup),( ),( xSxSSSd n x n ?= +∞?∞∈ 1= a /1 0
∞→n

 ?[ { }() n Sx (,)?∞+∞ dBá l ?b (ii)
?0)( =xS π A n 2 >
 )()(sup),( ],[ xSxSSSd n AAx n ?= ?∈ n A ≤ )(0 ∞→→ n
 ?[ { }() n Sx [,]A A? Bá l ?b
4
(i) 2 )( π =xS
 )()(sup),( )1,0( xSxSSSd n x n ?= ∈ 2 π = a /1 0
∞→n

 ?[ { }() n Sx dBá l ?b (0,1) (ii) 2 )( π =xS
 )()(sup),( ),1( xSxSSSd n x n ?= +∞∈ narctan 2 ?= π )(0 ∞→→ n
 ?[ { }() n Sx Bá l ?b (1, )+∞
5
 xxS =)(
?? n x n xxSxS n 11 )()( 2 2 ≤?+=?
? ^ )()(sup),( ),( xSxSSSd n x n ?= +∞?∞∈ )(0 ∞→→ n
 ?[ { }() n Sx (,)?∞+∞ Bá l ?b
6

 0)( =xS =? ) 1 () 1 ( n S n S n n n ) 1 1( ? a /1 0
∞→n

 ?[ { }() n Sx [0 dBá l ?b ,1]
7
(i)
??0)( =xS 0)0()0( =+?+ SS n
O 2 []=? )()( xSxS dx d n 0)ln1( 1 <+ n x n
 )2( ≥n ? ^ n n xSxSSSd n x n ln )()(sup),( )1,0( =?= ∈ )(0 ∞→→ n
 ?[ { }() n Sx Bá l ?b (0,1) (ii)
 0)( =xS =? )2()2( nSnS n 2ln2 a /1 0
∞→n

 ?[ { }() n Sx dBá l ?b (1, )+∞
8
(i)
 0)( =xS =??? ) 1 1() 1 1( n S n S n n n n n ) 1 1(1 ) 1 1( ?+ ? a /1 0
∞→n

 ?[ { }() n Sx dBá l ?b (0,1) (ii)
 1)( =xS =+?+ ) 1 1() 1 1( n S n S n 1 ) 1 1(1 ) 1 1( ? ++ + n n n n a /1 0
∞→n

 ?[ { }() n Sx dBá l ?b (1, )+∞
9
 ? ? ? ? ? ? ? ≠∈ = = 2 ],,0[0 2 1 )( π π π xx x xS
| ],0[ π∈ n x
 P¤ n x n 1 1sin ?=
5 2 π ≠ n x
 =? )()( nnn xSxS n n ) 1 1( ? a /1 0
∞→n

 ?[ { }() n Sx [0, ]π dBá l ?b
10
(i)
| ? ? ? << = = π π x x xS 01 ,00 )( ),0( π∈ n x
 P¤ n n x 2 1 sin =
5 3 =? )()( nnn xSxS 1 2 1 ? a /1 0
∞→n

 ?[ { }() n Sx (0, )π dBá l ?b (ii)
 1)( =xS )()(sup),( xSxSSSd nn ?= ],[x ?∈ δπδ δ n 1 sin1?= )(0 ∞→→ n
 ?[ { }() n Sx [, ]δ πδ? Bá l ?b
11
(i)
 x exS =)( =? )()( nSnS n nn e?2 a /1 0
∞→n

 ?[ { }() n Sx dBá l ?b (0, )+∞ (ii)
?? x exS =)( 0)0()0( =+?+ SS n
O?  sv H
 n []=? )()( xSxS dx d n 01 1 <? ? ? ? ? ? ? + ? x n e n x
 ? ^ )()(sup),( ],0( xSxSSSd n Ax n ?= ∈ n A n A e ? ? ? ? ? ? +?= 1 )(0 ∞→→ n
 ?[ { }() n Sx (0, ]A Bá l ?b
12
(i) x xS 2 1 )( =
 =? ) 1 () 1 ( n S n S n n ? ? ? ? ? ? ? 2 3 2 a /1 0
∞→n

 ?[ { }() n Sx dBá l ?b (0, )+∞ (ii) x xS 2 1 )( =
 S n (x) = ? ? ? ? ? ? ? ? ?+ x n xn 1 )( 2 1 1 1 xS x x n x =< ++ =
 4 ?? [] 0 4 1 ) 1 () 1 (2 1 )()( 2 3 >+ +++ ? =? x n xx n xx xSxS dx d n
 V? )()(sup),( ),[ xSxSSSd n x n ?= +∞∈δ )()( δδ SS n ?= δ δδ 2 11 + ? ? ? ? ? ? ? ? ?+?= n n )(0 ∞→→ n
 ?[ { }() n Sx [, )δ +∞ Bá l ?b 2. ! S n (x) = n( n x  n x 2 )
5f ??  {S (x)}  l ???Bá l ?
OK ?Ds ????D
' n ]1,0[ ∞→n lim ∫ 1 0 )(xS n dx ≠ ∫ ∞→ 1 0 lim n S n (x) dxb ￡ f ??  {S n (x)}  l ??]1,0[ 0)( =xS b| n x n 1 1?=
5 =? )()( nnn xSxS +∞→ ? ? ? ? ? ? ??? nn nn n 2 ) 1 1() 1 1(
 ?[ {S n (x)} dBá l ?b ]1,0[ ?? ∞→n lim ∫ 1 0 )(xS n dx ∞→ = n lim xxxn nn d)( 1 0 2 ∫ ? 2 1 =
 S ∫ ∞→ 1 0 lim n n (x) dx 0=
 ?[ dx ∞→n lim ∫ 1 0 )(xS n ≠ ∫ ∞→ 1 0 lim n S n (x) dxb 3. ! S n (x) = 22 1 xn x +
5 ò f ??  {S n (x)} ),( +∞?∞ Bá l ?
ó ? ? ? ? ? ? )( d d xS x n  ?Bá l ?
),( +∞?∞ ? K ?Dp? ????D
' ∞→n lim xd d S n (x) = xd d ∞→n lim S n (x) i?BM x∈ ? ?b ),( +∞?∞ 3
1
S n (x)= 22 1 xn x +
 0)( =xS
5 n xn x xSxS n 2 1 1 )()( 22 ≤ + =? )(0 ∞→→ n
 5 ?[ {S n (x)} ),( +∞?∞ Bá l ?b
2
 )(xS dx d n 222 22 )1( 1 xn xn + ? =
 )(lim)( xS dx d x n n ∞→ =σ ? ? ? ≠ = = 00 01 x x
 | n x n 2 1 =
5 )( nn xS dx d 25 12 )( =? n xσ a /1 0
∞→n

 ?[ ? ? ? ? ? ? )( d d xS x n  ?Bá l ?b ),( +∞?∞
3
?? 0=x )
 xd d ∞→n lim S n (x)
0= )(lim)( xS dx d x n n ∞→ =σ 1=
 ?[ )
 0=x ∞→n lim xd d S n (x) = xd d ∞→n lim S n (x) ?? ?b 4. ! S n (x) = n 1 arctan x n
5f ??  {S n (x)} ),0( +∞ Bá l ?
k ùK ?Dp? ????D
' ∞→n lim xd d S n (x) = xd d ∞→n lim S n (x) ^?? ? ? 3 S n (x)= n 1 arctan
 n x n n n x x xS 2 1 ' 1 )( + = ?
 =)(xS ∞→n lim S n (x) 0=
 0)(' =xS
 ?[ )1(' 2 1 )1(lim ' SS n n ≠= ∞→
' ∞→n lim xd d S n (x) = xd d ∞→n lim S n (x)  ?? ?b 1=x 5. ! S n (x) =
? a ^? ?bp a￥|′S?
 P¤f ??  {S nx xen ?α n (x)}  ]1,0[ ò Bá l ?
ó s ?DK ? V[?D
' ∞→n lim ∫ 1 0 )(xS n dx = S ∫ ∞→ 1 0 lim n n (x) dx
? p? ?DK ? V[?D
'BM x∈[0,1]? ? ∞→n lim xd d S n (x) = xd d ∞→n lim S n (x) b 6 3 (1) S=)(xS ∞→n lim n (x)
 7
¤?0= =)( ' xS n 0)1( =? ? nxen nxα n x 1 =
' =?= ∈ )()(sup),( ]1,0[ xSxSSSd n x n 11 ) 1 ( ?? = en n S n α
 ?[ 0),(lim = ∞→ SSd n n ?O?? 1<α H? ?
 ?[? 1<α H
 {S n (x)} ]1,0[ Bá l ?b (2) S ∫ ∞→ 1 0 lim n n (x)dx
 ∫ == 1 0 0)( dxxS ∫ = 1 0 )( dxxS n n e n nn ??? +? ) 1 1( 12 αα
 ?[?O?? 2<α H
? ? . ∞→n lim ∫ 1 0 )(xS n dx = S ∫ ∞→ 1 0 lim n n (x) dxb (3) xd d ∞→n lim S n (x) xd d = 0)( =xS
 xd d S n (x)
 )1( nxen nx ?= ?α ?? )1(lim nxe nx n ? ? ∞→ ? ? ? = ∈ = 01 ]1,0(0 x x
 ?[?O?? 0<α H
 ∞→n lim xd d S n (x)= xd d ∞→n lim S n (x) BM x∈[0,1]? ?b 6. ! S '(x)uW  ??
 ),( ba S n (x) = ? ? ? ? ? ? ? ? ? ? ? ? ? + )( 1 xS n xSn
 ￡ ü
 {S n (x)} =>Bá l ?? S '(x)b ),( ba 3 A? S ∞→n lim n (x)
 ?[o?￡ ü )(' xS= 0>?η , { })(xS n  []ηη ?+ ba , Bá l ?? b )(' xS | ηα <<0 , 5 )(' xS [ ]αα ?+ ba , Bá ?? , ' 0,0 >?>? δε , ∈? ",' xx [ ]αα ?+ ba , , o1 δ<? "' xx , ü? ? ε<? )"(')'(' xSxS b | ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? = αηδ 1 , 1 maxN , 5? ONn > ∈x [ ]ηη ?+ ba , H
μ ∈+ n x 1 [ ]αα ?+ ba , , 7 ? ^ =? )(')( xSxS n εξ <? )(')(' xSS
 ?[ {S n (x)} =>Bá l ?? S '(x)b ),( ba 7. !   ??
 7 )( 0 xS ],0[ a S n (x) = d t
 n = b ∫ ? x n tS 0 1 )(",2,1 ￡ ü
 {S n (x)} Bá l ?? 0b ],0[ a ￡ ! MxS ≤)( 0 , 5 MxdttSxS x ≤= ∫ 0 01 )()(
 ∫∫ ≤= xx MtdtdttSxS 00 12 )()( !2 2 x M=
 " ∫∫ = ? ≤= ? ? x nn x nn n x Mdt n t MdttSxS 0 1 0 1 !)!1( )()(
 " ?? !! n a M n x M nn ≤
 0) ! (lim = ∞→ n a M n n
 ?[ {S n (x)} Bá l ?? 0b ],0[ a 8. ! S(x)  ??
O S(1) = 0b￡ ü
 {x]1,0[ n S(x)} [0,1] Bá l ?b ￡ S(x)  ??
 ?[μ?
 !]1,0[ MxS ≤)( b ? , V? 0)1( =S 0,0 >?>? δε , [ ]1,1 δ?∈?x
? ? ε<)(xSx n b ?? { } n x  []δ?1,0 Bá l ?? ,
 V? ,N? , Nn >? [ ]δ?∈? 1,0x
? ? M x n ε <
 ? ^ ε<)(xSx n 8 BM ? ?
yN {x]1,0[∈x n S(x)} [0,1] Bá l ?b 9