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32

3.n! “n- ”

	n
n! = 1 2 … (n – 1) n = (n – 1)! n	n! = ∏k 0! = 1.
	k =1

! "

1.& , ! :

a)! «a0», «a1» ( -

! n=1);

b)a0 ( a1) ! x ;

c)! !.

2.' # ! +* :

a);

b)! ! . # ;

c)! ! ! .

$ , +* !

, !

*. :

an+1 = an Tn, Tn – . .

3. ! .

( # * lim an = 0 . !, -

n→∞

n, |an| > ε ! + -

! ε. $ n !

.

4., -

n !, -

Nmax ( , , 100). ' n > Nmax -

* * ! -

( an).

8 * !, :

a)|an| ε, , , «

» ! ! Nmax ;

b)|an| ε, ! .

. !:

−# x ;

− ! ε -

! Real – * # -

#; − , , .

#

sin x. .

% +*

33

∞

(

− 1 )

n

x

2 n +1

x

3

x

5

x

7

f ( x ) = sin x = ∑

= x −

+

−

+Κ ,

x

< ∞

( 2n + 1 )!

n =0

3!

5!

7!

*

an =

(− 1)n x 2 n +1

,

n = 0, 1, 2, ...

(2n + 1)!

' n = 0, a0 = x. / +* an+1 !

*!+

an+1 = an Tn,

n = 0, 1, 2, … .

/ . Tn

T

=

an+1

,

n = 0, 1, 2, … .

n

an

=

(− 1)n+1 x

2(n+1)+1

= (− 1)

n

(− 1)

x 2n

x 3

an+1

– подставлено n+1 вместо n

(2(n + 1)

+ 1)!

(2n

+

3)!

an =

(− 1)n x 2n+1

= (− 1)n

x 2n x

(2n + 1)!

(2n + 1)!

T

=

(− 1)n

(− 1)

x 2n x 3

: (− 1)n

x 2n x

=

(2n + 1)!

n

(2n + 3)!

(

1)n (

1)

x 2n x 3

(2n + 1)!

(2n + 1)! – подставлено an+1, an

=

−

−

=

(− 1) x 2

( )n

(2n + 3)!

(2n + 3)!

x 2n x

− 1

- ! , ! ! !

(2n + 1)! = 1 2 3 … (2n + 1)

(2n + 3)! = 1 2 3 … (2n + 1) (2n + 2) (2n + 3)

− x2

/ . Tn = ( )( )

2n + 2 2n + 3

		an +1 = an		− x 2
				(2n + 2)(2n + 3)

' n = 0 * an, a0 = x. ( a1 a2, !

+* :

n = 0

a1

= a0 T0

= x (− x2 )

1

= −

x3

= −

x3

2 3

2 3

3!

n = 1

a2

= a1 T1

= −

x3

(− x2 )

1

=

x5

=

x5

2 3

4

5 2

3 4 5 5!

# , -

. !.

34

!- " "

+ , -

.

. an ! #

a0 , a1 , … an , .

do while « |An| > eps», # N > Nmax.

		0
	F = sin x;	N = 0;	An = x;	Sum = An
	An, Sum, F		N (N = 0)
	/	| An | > eps
N = N + 1 (An "0")			(
			− x	x
!:	An = An (2 N + 2 ) (2 N + 3);
x, F, Sum, N, |F-Sum|	Sn = Sn + An ;		N = N + 1
/
		An, Sum, F N
		N >	Nmax
8 *

Stop

.3. 2-#

" " " "

Program Pro1 ! – –

! (%, ) & $ & &

Implicit None

Real:: x = 1.1	! x0
Real:: eps = 1E-04 !
Integer:: N	!
Integer:: Nmax=100	!
Real An, Sn, F	! A0,A1,A2..An, , %
Real Delta	! Sn F

!, :

Namelist /result/ x, F, Sn, n, Delta

!, :

Namelist /avaria/ x, An, Sn, F, N, Nmax

!-------------

35

Open(1,file=’Out.txt’)! %

Open(2,file=’An.txt’)! % AGrapher - An N Open(3,file=’Sn.txt’)! % AGrapherSn N

Open(4,file=’F.txt’)! % AGrapher - F ( N)

Write(1, *) ’% ! ’, eps

, ! do while, ! -#,

; ! -

. ( * !

Write(1,avaria)

! /,"01# 2,3.4014

Delta=ABS(F–Sn)! Sn % F Write(1,result) !

End Program Pro1

Pro1

1.Out.txt, # (eps x) -

F, Sn, n, Delta = ABS(F – Sn). 1 Delta < eps, ! !.

Delta > eps , #, -

+. . !:

− ! ε !-

* # # Real. , -

! ! Real*8; − + , .

2.An.txt, Sn.txt, F.txt . ' -

*!+ AGrapher, !,

lim An = 0	,	lim Sn = sin(x) .
n→∞		n→∞

Pro2, " "

( Pro2 ! , ! Pro1

! # , ! ..

1.' ! 10 – 15 x [xm , xk]

xh % ! x

Program Pro2 ! x

. . .

Real:: xm=-1.6, xk=1.6, xh=0.2 !

!!! 5 % AGrapher

Do x = xm, xk+xh/2, xh ! $

!*** x, Pro1

Write(1,*) ’x=’, x, ’ F=’, F, ’ Sn=’, Sn, & ’n=’, n, ’ |F-Sn|=’, Delta ! $

EndDo

End Program Pro2

36

! !, Delta eps;

Delta < eps # x, +* ..

2.& !

a)! Format

Write(1,12) x, F, Sum, N, Delta !

12 Format & ! x

(’|’,F6.1,2(’ |’, F7.4),’ |’,I7,4x,’ |’,e9.2,’ |’)

b)« » « » “!”

out.txt, (a), -

! ,

Write(1,10) eps ! $

10 Format(’% ! ’,E7.1 /& ! x

’+	-------+---------	+---------	+-----------	+----------	+’/&
’|	x | станд	| Тейлор	|Членов ряда| Разница		|’/&
’+-------	+---------	+---------	+-----------	+----------	+’)

11 Format &		! « » $
(’+-------	+	---------+---------	+-----------	+----------	+’)

Write(1,11) ! « » $

. f(x) -

ε; , ε = 10–3 -

! . !+ ! ,

ε = 10–5 – !+ . . ( 7 ).

$ ! %

1.- # * ? 4 # ( ). '-

.

2.' *

? / ε?

3.5 ? / . ? 5-

# # ! ?

4.5 ?

5.5 ? ' ! . -

. a11.

6.- ! , -

, ? 5 # #

+ ?

7.!+ # ! + do while?

8.!+ # ! + exit?

9.0 , +* ( «˜»

): «при˜х˜=˜0.05˜sin(x)˜=˜˜.85E-05»

37

10./ !

1 Format(’|’,F12.7, ’|’, F12.7, ’|’, F12.7, ’|’)

11.!

Real:: f; integer:: n write (1,7) f, n

7 Format (I6, 5x, ’|’, e11.4)

12./ Pro1 ! do while -

do? ' -# .

13.- ! # Pro2 -

ε, # ! #. & 3, .

# $& !

1.. # -

( , ! # ).

2.! -# do while

Out.txt a b = a2 − 2 , a = 8 -

+* a, * -

. ! a.

a)a b ! “E9.3” #.

b)' ! .

c)- a # ?

3./ ! ( )?

0 ?
Real:: f=2700000, z=-0.00017
write (*,4) f, z; write (*,	5) f, z
4 Format (F7.1);	5 Format (2E9.2)

4.0 Format, +* ( «˜»

):

˜:˜˜.68E-04˜:˜˜.75E-02˜:˜˜.13E-03˜:

% 8.

$' ! « & »

5 a0 x0 + Pro1.

7

8

nπ

x

< ∞

∞

xn cos a +

2

cos a

cos(x + a) =

∑

2

= cos a − x sin a−

x

+Κ

π

n!

2!

0 < a <

1

=

n 0

2

π

= − sinα

a0

= 1.5

cos α +

2

x0

= 1.5

38

7

8

6

x

< ∞

2

∞

x

2 n

x

2

x

4

x

6

Ch x = ∑

= 1 +

+

+

+Κ

0 = 3.5

( 2 n)!

n = 0

2!

4!

6!

x + 1

∞

1

1

1

1

>1

= 2∑

x

3

ln

= 2

+

+

+Κ

x −

( 2n + 1 )x 2n +1

3x 3

5

1

n = 0

x

5x

0 = 1.4

∞

n

n

2

3

4

4

e − x = ∑

( −

1 )

x

= 1 − x +

x

−

x

+

x

−Κ

x

< ∞

n!

n = 0

2!

3!

4!

0 = ±1.4

∞

1 3 5Κ ( 2n −1)

Arch x = ln2x − ∑

=

2 4

6Κ 2n 2n x

2n

=

n 1

x > 1

5

1

1 3

1 3 5

= ln2x −

+

+

+Κ

0 = 1.1

2

2 4 4 x

4

2 4 6

6 x

6

2 2 x

Archx = ln(x +

,-

x2 −1)

x

>1

∞

2n+1

arcsinx = x + ∑

1 3 Κ (2n −1) x

=

x

<1

2 4 Κ 2n (2n +1)

6

n=1

3

5

7

9

0 = 0.8

= x +

x

+

1 3 x

+

1 3 5 x

+

1 3 5 7 x

+Κ

2 3 2 4 5 2 4 6 7 2 4 6 8 9

∞

( −

1 )

n

x

n +1

x

2

x

3

x

4

− 1 < x ≤ 1

7

ln(x + 1 ) = ∑

= x −

+

−

+Κ

n + 1

n = 0

2

3

4

0 = ±0.6

∞

(x − 1)n+1

x − 1

(x − 1)2

(x − 1)3

x>

1

ln x = ∑

8

=

+

+

+Κ

2

(n

+ 1 )x n+1

x

2x 2

3x 3

n=0

0 = 0.6

π

∞

( −

1 )

n +1

x

2 n +1

π

x

3

x

5

arcctg x =

+ ∑

=

− x +

−

+Κ

2n + 1

x

≤1

9

2

n = 0

2

3

5

π

arcctg x=

− arctg x

0 = ±0.7

2

∞

( −1) n (2n)! x2n+1

2!x3

4! x5

6!x7

Arshx =∑

= x−

+

−

+Κ

< 1

4n

2

(2n

+1)

4 3

2

2

7

x

10

n=0

(n!)

42(2!)

5

43(3!)

0 = 0.95

,-

Arsh x =ln (x +

x2 +1)

x

<1

1 + x

∞

x

2 n +1

x

3

x

5

x

7

<1

11

ln

= 2 ∑

= 2

x +

+

+

+Κ

x

1 − x

n = 0

2 n + 1

3

5

7

0 = 0.8

39

7

8

nπ

x

< ∞

∞

xn sin a +

2

sin a

2

sin(x + a) =

∑

= sin a + x cos a −

x

+Κ

π

0 < a <

12

n=0

n!

2!

2

π

+

= cosα

sin α

a0 = 1.5

2

x0 = 1.5

∞

x

2 n + 1

x

3

x

5

x

7

Arth

x = ∑

= x +

+

+

+ Κ

2 n +

1

3

5

7

x

<1

13

n = 0

,-

Arthx =

1

ln

1 + x

x

<1

0 = 0.8

2

1 − x

∞

x

n

x

2

x

3

x

4

x

< ∞

14

e x = ∑

= 1 + x +

+

+

+ Κ

n = 0

n !

2!

3!

4!

0 = ±1.4

6

x

< ∞

15

∞

x

2 n +1

x

3

x

5

x

7

Sh x

= ∑

= x +

+

+

+Κ

0 = 3.5

( 2 n +

1 )!

n = 0

3!

5!

7!

∞

2n+1

x −1

3

5

x > 0

lnx = 2∑

(x −1)

(x −1)

(x −1)

16

= 2

+

+

+Κ

2n +1)(x+1)2n+1

n=0 (

x +1

3(x+1)3

5(x

+1)5

0 = 0.2

π

∞

( − 1 )

n +1

π

1

1

1

x > 1

17

arctg x =

+ ∑

=

−

+

−

+Κ

2n + 1 )x 2n +1

3x 2

5x 5

2

n =0 (

2

x

0 = 1.2

π

∞

(2n)!x2n+1

π

[ x +

2! x3

4! x5

6! x7

+Κ ]

x

< 1

18

arccos x =

−∑

=

−

+

+

2

2

+1)

2

42

2

5

2

0 = 0.8

n=0 4n (n!)

(2n

4 3

(2!)

43(3!) 7

∞

x n

x

2

x

3

x 4

− 1 ≤ x < 1

19

ln( 1 − x) = − ∑

= −

x

+

+

+

+Κ

n

2

3

4

n =1

0 = ±0.6

20

x

∞ (x ln a)n

(x ln a)2

(x ln a)3

x

< ∞

a

=

∑

n!

= 1 + x

ln a +

2!

+

3!

+Κ

a0 = 1.5

n=0

x0 = 3.5

∞

(

− 1 )

n

x

2 n

x

2

x

4

x

6

x

< ∞

21

cos x = ∑

= 1 −

+

−

+Κ

( 2 n)!

4!

n =

0

2!

6!

0 = 2.5

∞

(

−

1 )

n

x

2 n +1

x

3

x

5

x

7

≤1

22

arctg

x = ∑

= x −

+

−

+Κ

x

n = 0

( 2n + 1 )

3

5

7

0 = 0.8

e − x 2

∞

( − 1 )

n

x

2 n

x

4

x

6

x

8

x

< ∞

23

= ∑

= 1 − x 2 +

−

+

−Κ

2!

3!

n = 0

n!

4!

0 = 1.2

40

7

8

∞

( 2 n ) !

Arch

x = ln 2 x − ∑

=

4 n ( n !)

2

2 n x 2 n

n =1

x > 1

24

2!

4!

6!

= ln 2 x −

+

+

+ Κ

0 = 1.1

4 2 x 2

4 2 ( 2 !)2

4

x 4

43 (3!)2 6

x 6

- Archx = ln(x +

)

x2 −1

x

>1

∞

n

− 1)

n+1

(x − 1)

2

(x −

1)

3

0 < x≤2

25

lnx = ∑

( − 1)

(x

= (x − 1) −

+

+Κ

(n + 1)

0 = 0.3

n=0

2

3

0 = 1.7

sin x

∞

(

− 1 ) n x 2 n

x 2

x 4

x 6

x

< ∞

26

= ∑

= 1 −

+

−

+Κ

x

( 2 n

+ 1 )!

3!

5!

7!

60 = 4.5

n = 0

π

∞

( − 1)

n+1

π

1

1

1

x < −1

27

arctgx = −

+ ∑

= −

−

+

−

+Κ

2n + 1)x 2n+1

3x3

5x5

2

n=0

(

2

x

60 = -1.2

∞

1

1

1

1

Arcth x = ∑

=

+

+

+Κ

( 2 n + 1 )x 2 n +1

3 x 3

5 x 5

>1

28

n = 0

x

x

- Arcthx =

1

ln

1 + x

x

>1

60 = 1.3

2

x − 1

∞

( −1)

n

2n+1

Arshx = x + ∑

1 3 Κ (2n−1) x

2 4 Κ 2n (2n+1)

=

n 1

x

< 1

29

3

5

7

= x −

x

+

1 3

x

−

1 3 5

x

+Κ

60 = 0.95

2 3 2 4 5 2 4 6 7

Arsh x =ln (x +

,-

x2 +1)

x

<1

∞

( 2 n )! x 2 n +1

arcsin x = ∑

=

2

+

30

n = 0 4 n

( n !)

( 2 n

1)

x

<1

2 ! x 3

4 ! x 5

6 ! x 7

= x +

+

+

+Κ

4

3

4 2 ( 2 !)2

5

4

3

( 3!) 2 7

2.4. , !

0 ! , :

1.12 * # in.txt

. ! +* out.txt.

2.( ! , ! , -

out.txt.

41

3.! # #, # -

, .

4.' # ! # -

out.txt, ! .

! "

1.' # # # In.txt ! !

+ . ( !

, # + ! +

- # ! !, !,

! D . .

2.# ! out.txt !

.

3.1 - - -

# * # ., +*

* ( ).

4., ! .

, ! .

5.# !

# .

1.0 .

2., , .

3.% ! .

4.1 -# , c .

5..

6.out.txt.

% 9.

* # 1, 2, 3, …, n

A

=

a1 +a 2 +a 3 +... +a n

n

n

Gn

= n a1 a 2 a 3 ... a n

=

1

( 2

+

2

+

2 )

Qn

a2 ...

a1

an

n

M n

=

n

,

1

+

1

+

1

+ ... +

1

a1

a2

a3

an

ai ≠0,

n – # .