1) , , x:
2) :
3) :
.
.
(7.21)
.
, (-1, +1). (7.21),
S ¢(x) = 1 + x + x 2 +... + xn +....
- q = x, ½ x ½<1. , ,
C. (7.21) , S (0) = 0. ,
0 = - ln (1-0) + C, C = 0.
, S (x) = - ln (1- x) = .
( - ). (x-a), ..
C 0 + C 1(x-a) + C 2(x-a)2+... + Cn (x - a) n +... (7.22)
, y = (x - a) (7.18). , (7.18 , (7.22) , x = a.
.
. . (7.20)
-1 < ( 3 -1) < 1 0 < x 3 < 2.
, 0 < x < .
. f (x) = . ,
C 0 + C 1(x - a) + C 2(x - a)2+... + Cn (x - a) n + = (7.23)
f (x), ..
. (7.24)
. , f (a) = C 0. (7.24) =, C 1= f ¢(a). (7.24) = , C 2 = f¢¢(a)/2. (7.24) =, , (7.23)
f (x).
. f (x) , , (2.25) n ¥.
x =a
f (x) x=a ( (x-a)).
a=0
|
|
f (x) =
.
1. f (x) = ex.
ex =(ex)¢=(ex)¢¢ =...(ex)(n), x =0 f (0) = 1; f ¢(0) = 1; f ¢¢(0) = 1;...; f (n)(0) = 1;....
, y=ex x =0
ex = (7.25)
(7.25) y=ex.
2. f (x) = sin x.
. f (x) = sin x :
, , .
, sin x:
cos x:
cos x = .
.
p p.
4.1. .
( ), () . , , , , - , , . .
, A n . { a 1, a 2,..., ar } r A. . : , . . , : , . A. , , , - . a i m (a). .
, :
) ;
) .
.
( ):
m , n , m + n .
. , 2; b , 3. , b .
. (2; 4), ˗ (3), .
():
m , n , (, ) m n .
. 4 1, 2, 3, 4 2 W 1 W 2. , 1 W 2 , 4 W 1 . ?
: 4 2 2=6.
. 0, 1, 2, 3, 4, 5. ?
: .
. .
. n k (), n 1 , n 2,..., nk , .. n1 , n 2 .. , ? , n 1 ∙ n 2. , . . n 1 ∙ n2 ∙ n3. N k ,
|
|
N = n1 ∙ n2 ∙... nk. (1.1)
1. . , . 2, 3, 4 5, . ?
. , , .. n 1 = 3, - 2, 3, 4 5, n 2 = 4. 10 , .. n 3 = n 4 = n 5 = n 6 = 10. N = 3´4´10´10´10´10 = 120 000.
, k m,
N = m k (1.2)
2. . ? ( , 1 2 2 1 - , . . , , ).
. k = 2. , , , m = 6,
N = 62 = 36.
, , : , .
. n n , .
(1.3)
:
1- n ;
2- (n -1) ;
3- (n -2) ..
.
1. 5 ?
. 5=5!=120
2. . ?
. .
. : k n . n k k , () . k n :
(1.4)
1. ,, .
. 6 . ,,,,,.
2. , ?
. .
. n - k k-, ( ). , . k!
. (1.5)
: . : .
1. 10 . 3 ?
. .
2. 16 , . ?
|
|
. .
. - . , .
, . , . , ,
3: , , .
. , ( ) , .
1. . "" "". ( ), , , " " ( )
2. . , .. 1 6. , , 4.
. , . Ø.
. 8 .
. , .
. 1 6 .
. "" "" . "" " " "" , .
, , . .
. , .
, . . . ( -) .
. , .
1. , . .
2. 5. 3.
. 1, 2, , n , .
. 1 , 2 3 . . ; .
. 1, 2, , n , , .
. 1 6 ().
. , , (, ). , .
|
|
, ˗ 2 , 6 .
. , , .
, , 3, : 3 6 , .
. (), (, , ).
, , , .
(3.1)
. (). ,
.
, . , .
1. 10 . 3 ˗ , 2 , ˗ . , : ) , ) , ) .
, . , , .
, (10.8), :
2. . , , 10 , ?
. - , . 10 2: ; (1 2 , 2 3, . . 10 1)
.
: .
. : 10 , ˗ , ˗ ˗ . , . , , . , , . . . , , , . .
.
, , , , . 5:2:3 , .
(3.2)
-.
, (XVIII.) (XIX .), 4040 0,5069, 23000 () - 0,5005, .. , 0, 5.