.

( )

- C :

020804 , 020304

2005

2

-

2 2005 1

: .., ..

-

2 , : , .

3

-

, . .

1. 1.1. ,...,...,,, 321 naaaa

==+++++

1321 ......

nnn aaaaa (1)

. ,...,...,,, 321 naaaa , na

.

;a...aaaS;...;aaaS;aaS;aS nn ++++=++=+== 321321321211 (1).

, :

...,...,,, 321 nSSSS . (2) (1) ,

(2) S , (1)

......321 +++++= naaaaS

==

1nnaS (3)

, (1) . .

1.

= +1 )1(1

n nn.

nS - n -

.)1(1...

321

211

+++

+

=

nnSn

4

211

211

-=

, 31

21

321

-=

, 41

31

431

-=

, .

, ,1

11)1(

1+

-=+ nnnn

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

+

-++

-+

-+

-=

)1(11...

41

31

31

21

211

nnnSn = )1(

11+

-nn .

+

-= )n(nn

limSlimnnn 1

11=1.

2. ,

=

-- -=+-++-+-1

11 )1(...)1(...1111n

nn

: 11 =S , 02 =S , 13 =S , 04 =S , , , ,

. 3. , :,...,...,,,,1 132 -nqqqq

=

-- =+++++1

1132 ......1n

nn qqqqq .

1q , , ,

11...1

1132

--

=+++++=-

-

qqqqqqqS

nn

n ,

qq

qqqS

nn

n --

-=

--

=11

111

.

1. 1||

5

3. 1|| >q , =--

= q

qSlimn

nn 11

, . . .

1.2. (1) , m- ,

...,...321 +++++ ++++ kmmmm aaaa (3) m- (1). 1.1. (3) ( ) (1). . . . mr - m- . 2.1. mr m- (1) m . 3.1. (1) S,

=1nnca , c - , ,

.cS

4.1.

=1nna

=1nnb

S s , )(1

nn

n ba +

=

s+S .

5.1( ).

=1nna

, , .. 0=

nnalim .

. 4.

=

=+++++1

1...1...31

211

n nn. (4)

. , ,

6

01 == n

limalimnnn

(4). , S .

21

21

21

11

2 =>+++=-

nn

n...

n)SS(lim nnn .

, . (1) , 0>e N, n>N p

e

7

11. 123

2 +-

=nnan ; 12. n

n

nna

2)1(-

= ; 13. 2)1(2

na

n

n-+

= ;

14. nnn ))((a

131-+

= ; 15. !

cos2

sin2

n

nn

anp

p

+

= .

1.3.

6.1.

=1nna

, , . . 7.1( ). nn ba 0 , 0nn = ,

=

=+++++1

321 ......n

nn bbbbb (5)

, (1) . (1) , (2). 8.1( ). (1) , (5)

cbalim

n

nn

= ,

c=const, 0c , (5) (1), (1) (5) . ,

nn ba ~ , n , na nb .

5.

= +1 2)1(1

n n.

( . 1).

= +1 )1(1

n nn. ,

)1(1

)1(1

2 +a , 1a

. , 1>a 1a . , 0a ,

10

n , .. ( . 5).

, ( ): 1. ...)(... n +-++-+-+- -11111111

2. ...... +

++

+

+

n

n 521

52

31

52

21

52 32

3. ...nn... +

++

++++12

174

53

32

4. ...)(...

n

n

+-

+-+-+

+

1

1

43 101

101

101

101

5. ...n... +++++

21

61

41

21

6. ...n... +

+++++

1101

311

211

111

7. ...)n(n... +

+++

+

+

11

431

321

211

8. ...n...

n

+++++2

32

222

32

9. ...n... ++++++ 1

41

31

211

10. ...)n(... +

-++++ 2222 13

181

51

21

11. ...)1(...

343

232

21 333

++

++++nn

n

( ):

12. ...)(n...

n+

-++++

212

225

23

21

13. ...)n...()n...(... +

--

++

+

+3495113852

951852

5152

12

11

14. ...nn...

n

+

-+

++

+

+

121

54

33

12 32

15. ...nn...

n

+

-+

+

+

-1253

1283

52

21

:

1. ...!1...

!31

!211 +++++

n

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

151

81

31

2 +-+++++

n

3. ...)13()23(1...

1071

741

411

++-

++

+

+ nn

4. ...12...

199

94

31

2

2

++

++++nn

5. ...1...

103

52

21

2 ++++++

nn

6. ...)2()1(12...

547

435

323

22222222 ++++

++

+

+ nn

n

7. ( )...

133...

109

76

43

2

22

++

++

+

+

nn

8. ...1312...

107

75

43 22

321

+

++

++

++

n

nn

9. ......2781 3

32 +++++ nen

eee

10. ...122...

34

221

1

32 ++++++

-

n

n

11. ...12!...

12!3

12!2

12!1

32 ++++

++

++

+ nn

12. ...)!1(2...

!24

!121

1

+-

++++-

n

n

13. ...4...1284)12(...531...

1284531

8431

41

+

-++

+

+n

n

14. ...)!2()!(...

!6)!3(

!4)!2(

!2)!1( 2222

++++n

n

12

15. ...)n...()n...(... +

-+

++

+

+23741

3998100410021000741

10041002100041100210001000

16. ...)n)(n...()n)(n...(... +

----

++

+78118171395146761411852

951852

12

17. ...)n)(n(...)n(...... +

---

++

+

244410864234951

64251

21

18. ...)!12()910(21111...

!521111

!3111

!11

+-

-++

+

+

nn

19. ...)12(...97531...941...

97531941

531411

2

+-

++

+

+

nn

20. ( ) ...212...

225

23

211 +-+++++ n

n

21.

=1

1n n

arcsin ; 22.

=12

1n n

sin ;

23.

=

+

1

11n n

ln ; 24.

=

+1

2

2 1n n

nln ;

25.

=2

1n nln

; 26.

= 21

n nlnn;

27.

= 2 21

n nlnn; 28.

= 21

n nlnlnnlnn; 29.

= -2 21

n nn;

30.

= +1 11

n )n(n; 31.

= ++1 211

n )n)(n(n;

32.

= +2 31

n nlnnlnn; 33.

= -2 31

n nnn;

34.

= --1 33

1512n )n()n(n

;

35.

=

-

1cos1

n np

; 36.

=1

!n

nnn

; 37.

=1

!2n

n

n

nn

;

38.

=1

!3n

n

n

nn

; 39.

=1

!n

n

n

nne

; 40.

=1

!5n

n

n

nn

13

1.4. , . , .

...)1(... 14321 +-++-+-+

nn aaaaa , (7)

0>na . 12.1( ).

(6) : ......321 >>>>> naaaa : 0=

nn

alim , .

12.

=

++ -=+-++-+-1

11 1)1(...1)1(...41

31

211

n

nn

nn ,

, . . :

1) >>>> ...31

211 ; 2) 0

1 = n

limn

.

. . -

=

=+++++1

321 ......n

nn aaaaa , (8)

...,...,,, 321 naaaa , , . , (8):

=

=+++++1

321 ||...||...||||||n

nn aaaaa (9) 13.1( ). (9), (8). 1. (8) , (9) . (8) , (9) , (8) . (8) . , (8) ,

14

11 +

|a

a|limn

nn

1>

nnn

|a|lim ,

(8), (9). rn 1+ nn b|r| . 13.

...12

)1(...74

53

321 2

)1(432

+

--++

+

-

-

- nnn

nn

.

:

...12

...74

53

321

432

+

-++

+

+

+

n

nn

. .

,

n

limnnlim

nnlim

nnn

n

n 21

12

11212

=-

=-

=

-

. , 12, ( ), . .

...1...

31

211 +++++

n - ( ).

. .

1. ...12)1(...

51

311

1

+-

-+-+-

-

n

n

2. ...)1(...

31

211

1

+-

+++--

n

n

3. ...)1(...

91

411 2

1

+-

+-+--

n

n

15

4. ...56)1(...

133

721

1

+-

-+-+-

-

nnn

5. ...)1(12)1(...

437

325

213 1 +

++

-+-

+

-

-

nnnn

6. ...2)1(...

83

42

21 2

2

+-+-+--+

n

nn n

7. ...n)n(n)(... n +

-+++

-++-

--

+-111

11144

4133

322

2

8. ...1312)1(...

107

75

43 32

+

++

-++

-

+-

nn

nn

9. ...)13(...852)12(...753)1(...

852753

5253

23 1 +

-+

-+-

+

- -nnn

10. ...)52(...1197)23(...741)1(...

1197741

9741

71 1 +

+-

-+-

+

- -n

nn

11. ...)10(lnsin...

)10(ln3sin

)10(ln2sin

10lnsin

32 ++-++ nnaaaa

12.

=

-

1

1n

n

nnln)(

; 13.

=

--1

1 11n

n

nntg)( .

2. . 2.1. , ,

: ...)(...)()()( 321 +++++ xuxuxuxu n (1) . ,

......1 332 ++++++ xxxx . (1) x - 0x )(xun , ,...3,2,1=n , ...)(...)()()( 0030201 +++++ xuxuxuxu n (2) . , 0x (1). (2) , 0x

16

(1). , )(xun , (1) , - .

2.1. .

)(xS (1) x , (1).

=)(xS ...)(...)()()( 0030201 +++++ xuxuxuxu n n ( n - )

)(xSn , )(xrn .

=)(xSn ...)(...)()()( 0030201 +++++ xuxuxuxu n )()()( xSxSxr nn -=

, x )(xSn n . , ,

)(xSn . , x ,

)x(S)x(Slim nn = , 0= )x(rlim nn (1)

, x . 1.

...2

)1(...23

)1(22

)1(211

3

3

2

2

++

+++

++

++

n

n

nxxxx

)(xun , :

.|x|

|x|)n(n|x|lim

|)x(u||)x(u|lim nn

nn

nn

n

n 21

11221

1

11 +=

+++

=+

+

+

,

( ), 12

|1| , (4) 0=x . , , , R, x, : Rx || , . 2.3. (4) R , x, Rx || , . ),( RR- . (3)

),( 00 RxRx +- .

2.2. ,|a||a|

limn

n

n01 +

(4) =R |a||a|lim

n

n

n1+

.

2.

...!

1...!2

11 2 +++++ nxn

xx

=R |a||a|lim

n

n

n1+

= =

+ !n

)!n(limn

1=+

)n(lim

n1 .

, .

3.

=1!

n

nxn ,

0=x ,

19

=R|a||a|lim

n

n

n1+

= =

+ )!n(!nlim

n 10

11

=+ )n(

limn .

4.

=1

||n

n

nx

.

=R =+

|a||a|lim

n

n

n1

=+

nnlim

n

1 111 =+

)n

(limn

.

, 2.2 (-1,1). , ..

1-=x , 1=x . 1=x

=1

1n n

,

1-=x ,

=

-1

1)1(n

n

n , .

, [ )1,1- .

:

1.

=0n

nx . 2.

= 1 2n nn

nx

.

3.

=

-

-1

12

12n

n

nx

. 4.

=

--

-12

121

)34(2

n

nn

nx

.

5.

=

--

1

1)1(n

nn

nx

. 6.

= ++

1

25

12)1(

n

n

nxn

.

7.

=

- +-1

21 )12()1(n

nn xn . 8.

=1 !n

n

nx

.

9.

=1!

n

nxn 10.

=1nn

n

nx

.

11. n

n

nx

nn 12

1 12

-

=

+ . 12.

=0

223

n

nn x .

20

13.

22

1 21

+

=

xn

nn

. 14.

=1

!n

n

n

nxn

.

15.

=

-

2

1

3n nn

nlnnx

. 16.

=

-

-

-1

1

351

nn

nn

n)x()(

17.

= -

1 5)3(

nn

n

nx

. 18.

= -

1

2

9)1(

nn

n

nx

.

19.

=

- --1

21

2)2()1(

n

nn

nx

. 20.

=

+

12

)3(n

n

nx

.

21.

=

+1

)3(n

nn xn . 22.

=

-

+

1

12

42)5(

nn

n

nx

.

23.

= --

1 2)12()2(

nn

n

nx

. 24.

= ++-

1

2

)1ln()1()3(

n

n

nnx

.

25.

=++

--

112 2)1()3)(23(

nn

n

nxn

. 26.

= ++-

-1 1)12(

)3()1(n

nn

nnx

.

2.3. )(xf ...,...)( 2210 +++++=

nn xaxaxaaxf (5)

),( RR- . , ),( RR- )(xf ( x). 2.3. )(xf ),( RR- (5), )(' xf (5), .. ...,xna...xaxaa...)'xa...xaxaa()x('f nn

nn +++++=+++++=

-12321

2210 32

)(xf . ,

(5). 2.4. )(xf ),( RR-

(5), ),( RR-

21

(5), .., ),(, 21 RRxx - ,

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

1

2

1

2

1

2

1

2

1

102

210 ++++=+++++= x

x

nn

x

x

x

x

x

x

nn

x

x

dxxaxdxadxadxxaxaxaadxxf

,

:

1. ...nx...xxx

n

+++++32

32

2. ...nx)(...xxx

nn +-+-+- -1

32

132

3. ...nx...xxx

n

+-

++++-

1253

1253

4. ...nx)(...xxx

nn +

--+-+-

--

121

53

121

53

5. ...x)n(...xx n ++++++ 1321 2 6. ...x)n()(...xx nn +--+-+- -- 22142 121531 7. ...x)n(n...xx n ++++++ -12 1433221 :

8. ...xn...

xxx n+++++ 32

321

9. ...nx...xxx

n

+-

++++-

3495

3495

2.4.

2.5. )(xf ),( 00 RxRx +-

=)(xf ...)(...)()()( 03

032

02010 +-++-+-+-+n

n xxaxxaxxaxxaa , (6) .

22

...,)(!

)(...)(

!2)(''

)(!1

)(')()( 0

0)(

20

00

00 +-++-+-+=

nn

xxn

xfxxxfxxxfxfxf (7)

)(xf ,

),( 00 xfa = ,!1)(' 0

1xfa = ,

!2)('' 0

2xfa = , ,

!)( 0

)(

nxfa

n

n =

)(xf x . , )(xf 0xx - , . ,00 =x , :

...!)0(...

!2)0(''

!1)0(')0()(

)(2 +++++= n

nx

nfxfxffxf (8)

. , )(xf . )(xf ),( 00 RxRx +- . x n

),(...)(!

)(...)(!2

)('')(!1

)(')(

)(

00

)(2

00

00

0 xrxxnxfxxxfxxxfxf

xf

nn

n

++-++-+-+=

=

(9)

)()!1())((

)( 000

)1(

xxn

xxxfxr

n

n -+-+

=+ q

(10)

),(...!

)0(...!2

)0(''!1

)0(')0()()(

2 xRxn

fxfxffxf nn

n

++++++= (11)

23

1

)1(

)!1()()( +

+

+= n

n

n xnfxR x , ,xqx = 10

24

, n , , 0=

)x(Rlim nn x , ,

xe (14). 6. xxf sin)( = .

),kxsin()x(f )k(2p

+= 02

0 == pksin)(f )k( nk 2= , nkf )1()0()( -= 12 += nk

1|)(| )( xf n . xsin :

...)!(

)(...!

sin ++

-++-=+

121

3

123

nxxxx

nn

,

),( -x xcos :

...)!(

)(...!

cos +-++-=n

xxxn

n

21

21

22

x ,

:

1. )0(, >aa x . 2. ).4sin( p+x

3. )cos( ax + . 4. x2sin . 5. )2ln( x+ . x

: 6. x2cos . 7. .3cos3sin xxx +

8. 2)1(32

--

xx

. 9. 34

532 +-

-xx

x.

10. .2xxe- 11. .2xe

12. .2cos x 13. .9 2x

x+

25

14. .4

12x-

15. .11ln

xx

-+

16. ).21ln( 2xx -+

, x , :

17. )1ln()1( xx ++ . 18. arctgx .

19. .arcsinx 20. )1ln( 2xx ++ .

, x , :

21. .cossin 22 xx 22. .)1( xex -+ 23. .)1( 3xe+ 24. 3 8 x+ .

25. 411x-

. 26. ).23ln( 2 ++ xx

x .

27. tgx . 28. xe cos .

29. .cosln x 30. .sin xe x 31. xln .1-x

32. x1

.1-x

33. 21x

.1+x

34. 23

12 ++ xx

.4+x

35. 74

12 ++ xx

.2+x

36. xe .2+x

37. x .4-x 38. xcos .

2p

-x

26

39. x2cos .4p

-x

40. xln .11

xx

+-

41. ,

!41

!31

!212 +++t ?

42. 4p ,

...53

53

-+-=xxxarctgx ,

1=x ? 43.

...,!2

1cos2

+-=xx

o18cos 0,001? 44.

...,!3

sin3

+-=xxx

o15sin 0,0001? 45.

...,!3!2!1

132

++++=xxxe x

e 0,0001? 46.

...,xx)xln( +-=+2

12

2ln 0,01, 0,001? 47. 3 7 0,01 3 8 x+ x . 48. 4 19 0,001.

49. x

...,!2

1cos2

+-xx

, 0,01, 0,001, 0,0001? 50. x

,xxsin , 0,01, 0,001?

27

3. 3.1

3.1. 3.1.

)sincos(2

...sincos...2sin2cossincos2

1

0

22110

nxbnxaa

nxbnxaxbxaxbxaa

nn

n

nn

=++=

=++++++++

(1)

; ,...,,...,,,,, 22110 nn bababaa - .

3.1. )(xf [ ]pp ,- ,

=

++=1

0 )sincos(2

)(n

nn nxbnxaaxf , (2)

, .

nn baa ,,0 :

dxxfa -

=p

pp)(10 . (3)

-

=p

ppnxdxxfan cos)(

1. (4)

-

=p

ppnxdxxfbn sin)(

1. (5)

3.2. )(xf - , [ ]pp ,- . ,,,0 nn baa (3)-(5), ,

=++

1

0 )sincos(2 n

nn nxbnxaa

)(xf . )(xf , [ ]pp ,- , , :

28

0=nb , dxxfa =p

p 00 )(

2, =

p

p 0cos)(2 nxdxxfan . (6)

)(xf , [ ]pp ,- , , :

0=na , =p

p 0sin)(2 nxdxxfbn . (7)

, )(xf , , , )(xf .

1. xxf =)( . , (7).

0=na ,

nnxdx

nnxx

nnxdxx

nxdxxfb

n

n

2)1(cos1cos12sin2

sin)(2

1

00

0

+-=

+-==

==

pp

p

pp

p

,

+-++-+-= + ...sin)1(...

44sin

33sin

22sin

1sin2 1

nnxxxxxx n .

2l )(xf [ ]ll,- (l-

) . (2)

=

++=1

0 )sincos(2

)(n

nn xlnbx

lnaaxf pp , (8)

dxxfl

al

l-

= )(10 . (9)

-

=l

ln xdxl

nxfl

a pcos)(1 , ,....3,2,1=n (10)

29

-

=l

ln xdxl

nxfl

b psin)(1 , ,....3,2,1=n (11)

[ ]pp ,-

1. ||)( xxf = . 5. xxf sin)( = . 2. xxf += p)( . 6. xxf cos)( = . 3.

2)( xxf = . 7. axxf sin)( = . 4.

xexf =)( . 8. axxf cos)( = . 9.

axexf =)( . 10. shxxf =)( . 11. chxxf =)( . : 12. ||)( xxf = , )11( - x 13. xxf 2)( = , )10( x 14.

xexf =)( , )( lxl - 15. xxf -=10)( , )155( - x : ) ; )

: 16. 1)( =xf , )10( x . 17. xxf =)( , )0( lx . 18.

2)( xxf = , )20( p x .

19.

30

20.

3

23 ,

31

:

.