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()
7 2013 .
1
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I 6
1 6
2 7
3 7
4 7
II 9
5 9
6 9
7 10
8 11
III 11
9 11
10 12
11 12
12 13
IV 13
13 13
14 15
15 15
V 16
16 16
17 16
18 16
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19 17
VI 18
20 18
21 19
22 - 19
23 .. 21
VII 21
24 21
25 22
26 22
VIII , - 23
27 23
28 24
IX 24
29 2429.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2429.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
30 25
31 26
X , 27
32 27
33 28
34 28
XI 29
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35 29
36 3036.1 . . . . . . . . . . . . . . . . . . . . . . . . . 31
37 32
37.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3237.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3337.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
XII 35
38 35
39 3639.1 . . . . . . . . . . . . . 36
39.2 . . . . . . . . . . . . . . . . . . . . . . . . . 36
40 36
41 37
XIII 37
42 3842.1 , . . . . . . . . . 38
42.2 . . . . . . . . . . . . . . . . . . 38
43 3843.1 0 . . . . . . . . . . . 3843.2 . . . 3843.3 . . . . . . . . . . . . . . . . . . . . 39
XIV 3943.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3943.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
XV 40
44 4044.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4044.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4144.3 . . . . . . . . . . . . . . . . . . . . . 41
45 41
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XVI I 4245.1 , . . . . . . . . . . . . . . . . . . . . . . . . . . 44
XVII II 44
XVIII III 46
XIX 47
XX 49
46 49
47 49
5
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I
1
- X Y
X Y ={(x, y)|xX, yY}
() R , :
1. RR +
R
(a) 0, x R x+ 0 = 0 +x= x(b)x R x: x+ (x) = (x) +x= 0(c) x,y,z R (x+y) +z=x+ (y+z)(d) x, y R x+y = y +x
1a, 1b, 1c G, G (). 1d, G ().
2. R
R
R
(a) 1, x R x 1 = 1 x= x.(b)x R \ {0} x1 :x x1 =x1 x= 1.(c) x,y,z R (x y) z= x (y z)(d) x, y R x y = y x
2a, 2b, 2c, G . 2d, .
1.+ 2. x,y,z R (x+y) z= xz+yz3.
x y R x y
(a)x R x x(b)x, y R, (x y) (y x), x= y .(c) : x,y,z R (x y) (y z)x z(d)x, y R (x y) (y x).
3a, 3b, 3c G, G -
. 3d, .
1.+ 3.x,y,z R, x yx+z y+z
6
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2.+ 3.x, y: 0 x, 0 y0 x y P 1, 2, P (). 3, P .
4. ()
X R Y R :x, y: x
ycR :x
c
y., X= [0;
2), Y = (
2;2] c Q.
2 -
,
a0, a1, a2, . . . , an, . . .
,a0 ,a1a2 . . . an . . . - ( {0, 1, . . . 9}). , +0, 00 . . . 0, 00 . . . , a0, a1 . . . an999 . . . a0, a1 . . . (an+ 1)000 . . . (an= 9) ( ).
3
P Q , -
() f: PQ, , ..x, yPf(x+y) =f(x) +f(y)
f(xy) =f(x) f(y),
f . x y f(x) f(y), f .
3.1. .
4 , X Y,
X Y, ..xX yY, X Y y Y X.
.
f, x1, x2X :x1=x2f(x1)=f(x2)f, yYxX :y = f(x)
- . .
7
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, X, X, . card X, |X|.
, . .
, X Y |X| |A|. [:|||||:]
X , N, .. card X =cardN =0 (-). X , , .
4.2.
R , -.
4.3 (). R , N.
. (0, 1). R, .. card(0, 1) = cardR. n :
0, a11a12 . . . a1n . . .
0, a21a22 . . . a2n . . .
. . . . . .
0, an1an2 . . . ann . . .
0, {a11, 0, 9}{a22, 0, 9} . . . {ann, 0, 9}. 0 1, n , .. 1- , 2- .. , , , .
[:|||||:]
4.4. ,
N.
8
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, - (0, 1). a = a1a2a3 . . . an. a f(a) =
ni=1 ai 2i. -
, (0, 1) 2.
II
5
MR (), A(B)R :xM, x A(B x).M R , C R :xM, |x| C., A M( ), B M (
).aX () X, xX, x
a (x a). , () , , .
() - sup X. () infX.
= sup X xX, x
< xX :x >
= infX xX, x > xX :x <
6
6.1 ( ). , - .
. X R Y ={y R| x X, x y} ( ).X
= Y
= .
R :xX yYx y. X Y.
X Y Y = min Y= sup X. [:|||||:]
6.2 ( ). , - .
. X R Y ={y R| x X, y x} ( ).X= Y= .
R :xX yYy x. X Y. X Y Y = max Y= infX. [:|||||:]
9
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X , sup X. sup X, .
: X = (0, 1), sup X = 1, infX = 0. , -, .
7 7.1 (). a n, 1 n , a.
1 + 1 + + 1 n
> a
7.2 (). a, n, n > a, ..a R n N :n > a.
. 7.1. [:|||||:]
7.3 ( 7.2). a b, 0 < a < b, k, (k 1)a b < ka.
.
(k 1)a b < kaka a b < kak 1 b
a< k
7.2 , n, b/a < n. k = min{n|n > b/a}. k
1 b/a. [:|||||:]
7.4. a b (a < b) . c, a b.
. a= 0, a1a2 . . . an . . . , b= 0, b1b2 . . . bn . . . . , a b . , , . i, ai < bi( , .. a < b). 2 :
1. bi> ai+1. c = 0, a1a2 . . . (ai+1)000 . . . . , , a b.
2. bi=ai+ 1. 2 :
(a) b = 0, b1b2 . . . bi . . . 0 . . . . j. c = 0, b1b2 . . . bi000 . . . 100 . . . , 1 j+ 1. a b . , - 1, a = 0, a1a2 . . . ai999 . . . , c= a.
(b) b = 0, b1b2 . . . bi000 . . . . , a= 0, a1a2 . . . ai999 . . . , .. - a = b. j > i, aj < 9. c = 0, a1a2 . . . ai . . . (aj + 1)1000 . . . . , b i- a j-. aj+1 = 1, ..
a= 0, a1a2 . . . ai999 . . . aj9999 . . . , c= a.
[:|||||:]
10
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8
8.1. n, :
1. n= 1.
2. - k = n n= k + 1.
. . , n. m, :
1. n= m .
2. n, m, ( , m , ).
, m > 1, .. n = 1 ( 1). -
, m1 . , m1 , m . - 2. [:|||||:]
III
9
NX x1, x2, . . . , xn.N R .
(), MR (mR) :n N xn M(xn m). ..{xn} , AR :n N |xn| A.
..{xn} , {xn} . ..{xn} , AR n N :|xn|> A.
{xn} ,
AR N(A) :n N |xn|> A
.... , , , .
xn=n+ (1)n+1 n
2A0 R :Nn0 N |xn0|< A0 ( ....)
A0 = 1 :N N k0 N :|x2k+1|= 0< 1 =A0{xn} ,
>0
N() :
n N()
|xn
|<
11
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10 -
{n} ....1.
{n
}.
. , > 0N() :n N |xn| < . , n N . n < N xn max{|1|, |2|, . . . , |N1}. = 1, A=max{|1|, . . . , |N1|, 1} n N, |n| A. [:|||||:]
2. {yn} , {n yn} .
.
>0N() :n N() |n|< A
A:n N yn A{n yn}: >0N() :n N |n yn|<
A A=
[:|||||:]
3. {n} .., {n n} {n n} .
.
>0N1() :n
N1 |n|0
N() :
n N
|xn
|<
1
|xn|>
1
=A
[:|||||:]
12
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11.2. {xn} .. n N, xn= 0,
1xn
} .
.
A >0N(A) :n N |xn|> A 1|xn| < 1
A=
[:|||||:]
12
a R , U(a).U(a) = (a; a + ) -. U (a) = (a; a)(a; a+) =
0
U= U \ {a} . , ..{xn} , a,
limnxn=a U(a)N(U(a)) :n NxnU(a) (1)limn
xn=a >0N() :n N |xn a|< (2)
12.1. .
. 1- . xnU(a), a < xn< a + 0N() :n N |xn a|< a < xn < a+
L >0 :
n
N
|xn
| L= max
{|x1
|, . . . ,
|xN1
|,
|a+
|,
|a
|}
L xn, n, n < N. [:|||||:]
3. limn
xn = a, limn
yn = b, limn
(xn yn) = a b, limn
(xnyn) = ab,
limn
xnyn
= ab
, b= 0.
.
limn
xn=a, limn
yn=b
xn= a+n, yn= b+n,
{n
},
{n
} ..
xn yn = (a+n) (b+n) = (a b) + (n n) ..
xnyn= (a+n)(b+n) =ab+ (nb+an+nn) ..
{ xnyn
ab} , .
:
13.1.
limn
yn=b
= 0.
r >0,
N
N :
n N
|yn
|> r >0
.
>0N() :n N |yn b|< b < yn< b+=
b
2r < b
2< yn 0N1 :n N1 |xn a|< 12 > 0N2 :n N2 |yn b|< 2
n N1 |xn a|< c a n N2 |yn b|< b cn max{N1; N2} xn< c a+a= c = c b+b < yn
[:|||||:]
14.2. limn
xn = a, limn
yn=b. :
N :n Nxn> yn, a bN :n Nxn yn, a bN :n Nxn> b, a bN :n Nxn b, a b
. 4 . , a < b. 14.1N :n N xn < yn ( -) xn < b ( ). . [:|||||:]
14.3 ( ). limn
xn =a, N :n Nxn[p, q], a[p, q].
. 14.2: xn p, a p. ,xn qa q. [:|||||:]
15
15.1.
N :n Nxn yn zn. limn
xn=a = limn
zn. limn
yn=a
.
>0N1 :n N1 |xn a|< a < xn< a+ >0N2 :n N2 |zn a|< a < zn < a+
a < xn yn zn< a+ limn
yn = a
[:|||||:]
15
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V
16
n N xn xn+1(xn xn+1), .. () {xn} (). .
n N xn > xn+1 (xn < xn+1), .. () {xn} (). -.
17
17.1. .. ....,
.
. - - 2. .
{xn} , {xn} . sup{xn}= S >0N :S < xN
{xn} n N xn xNS < xN xn S < S+ lim
nxn = S= sup xn.
[:|||||:]
17.2. .. - ...., .
. - - 2. .
{xn} , {xn} . inf{xn}= S >0N :S+ > xN
{xn} n N xn xNS+ > xN xn S > S lim
nxn = S= infxn.
[:|||||:]
18
18.1. en=(
1 + 1n
)n, yn=
(1 + 1
n
)n+1. lim
nen = lim
nyn.
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. yn.
yn1yn
=
1 +
1
n 1
n
1 +1
nn+1 =
n
n 1
n
n+ 1n n n
n+ 1=
n2
n2 1n
nn+ 1
(1 +)n 1 +n : n2
n2 1n
=
1 +
1
n2 1n
1 + n
n2 1 >1 +1
n=
n+ 1
n
n2
n2 1n
nn+ 1
>n+ 1
n n
n+ 1= 1
..n N yn1 > yn {yn}
1 +1
nn+1
= 1 +1
nn
1 +1
n> 2 1 +1
n= 2n+ 1
n > 2 ..{yn} .,{yn} -.
{en} ( ).
0< yn en =
1 +1
n
n1 +
1
n 1
0 n N
17
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. 1..
{an} , an b1 {an} {bn} , bn a1 {bn}
- 2
bn
an
0
limn
bn = limn
an= c
an c bn n N ( inf{bn} sup{an})cn=1
[an; bn]
[:|||||:]
19.1 . ,
n=1
0;
1
n
= 0
VI
20
n1 < n2 < . . . . {xnk} -{xn}. ,{1, 3, 5, . . . } / - N, {3, 1, 5, . . . } /N, .. n1> n2. 20.1. {xn} , -, .
.
limn
xn=a >0N() :n N() |xn a|< nk Knk N |xnk a|< lim
nxnk =a
[:|||||:]
20.2. / .. .
.
AN(A) :n N |xn|> Ank Knk N |xnk |> A {xnk} ..
[:|||||:]
18
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21
a ( ){xn}, -:
1. a {xn}
2. /{xnk} ..{xn}, a . 21.1. .
.
1 2 - a {xn}. - a, 1, 1
n, 2n
, . . . , 1n
, . . . . xk1 c -k1, xk2 k2, k2 > k1 .... , .. - a -{xn}. xk1 , xk2 , . . . , xkn, . . . , - a, ..|xkn a|< 1n .
2 1, {xn} , a. - a (, ). , - a {xn}.
[:|||||:]
21.2. - .. , ..
. .. , - 2 . 20.1.
[:|||||:]
22 -
22.1. .. /..{xn} . [a; b] :n N a xn b. [a; b] 2 . ( [a1; b1]) {xn}.
[a1; b1] {xn}. xn1 . [a1; b1] 2 . {xn}. [a2; b2]. xn2[a2; b2]. . xnk , k- , .. xnk[ak; bk].
{[ak; bk]} ..., 19.1!ck=1
[ak; bk]
limkak = limkbk =c limkxnk =c ( )[:|||||:]
19
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() {xn} () -.
limn
xn= max{k|k }limn
xn= min{k|k }
22.2. .. .
. - ( - ).xn . m, M :n N m xn M.{x} ={x| x ..{xn}}. , -{x} x, x M. , -{x} ( c, c m). , inf{x}. , x= inf{x} {xn}.
1. , x . > 0 . x = inf{x} y : y < x y / {x}. , x {xn}.
x= inf{x} >0x : x x < x+
{x} x , {xn}. x , x {xn}, (x , x) ( (x ,x+)) {xn}, .. x .
2. , x > x {xn}. = (x x)/2, (x ,x + ) (x ,x + ) = . - x x+. , > 0 x+ {xn}. , - x , {xn}, , x {xn}.
[:|||||:]
22.3. ..{xn} limn
xn = x, limn
xn = x. > 0 (x ,x+) {xn}.
. , x+ 2
x 2
{xn}. , x + 2 {xn} - 22.2. - . [:|||||:]
{xn}- +()(), M >0N(M) :n Nxn > M(xn M). 22.4. .. - /, /, - .
. .{xn} . Knk : xnk > K. nk < nk+1, N = nk n N xn > K
{xn} . [:|||||:]
20
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23 ..
23.1. .. - ...., -.
.
{xn} / - , .. lim
nxn = lim
nxn.
limn
xn = limn
xn = x. > 0 (x , x+) ... ,{xn} , ..{xn} -.
[:|||||:]
23.2. .. - ...., - /.
. {xkn}-{xk}. {xk}, .
{xk}. {xkn}. {xkn} {xk}. , , , {xkn}. [:|||||:]
VII
24
{xn} - ,
>0N() :n N, pN |xn+p xn|<
24.1. {xn} , N, - (xN , xN+) ..
>0N :n N, p N |xn xn+p|< |xN xN+p|<
xN < xN+p < xN+.. xN xN+ , ..
. [:|||||:]
24.2. -.
21
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.
>0N() :n N, pN |xn+p xn|< p N xN < xN+p < xN+
{xn} ., A >0 :n N |xn| A
A= max{|xN |, |xN+|, |x1|, . . . , |xN|}[:|||||:]
25
25.1. {xn} ...., -.
.
{xn} a: >0N() :n N()|xn a|<
2
p N |xn+p a|< 2
|xn xn+p|=|xn a+a xn+p| |xn a| + |xn+p a| {xn}
{xn} . {xn} .. 22.3 > 0 (x; x) (xn; xn+), , 0 x x
8/13/2019 Matan 8dec
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xn =n=1
an=a1+a2+a3
a2+a2+ a4+a5+a6+a7
a4+a4+a4+a4+ +a2n+ a2n+1+ +a2n+11+. . .
a2n+a2n++a2n
a1+ 2a2+ 4a4+ + 2na2n+ =n=0
2na2n
, xn xn+1, ..{xn} . , 17.1 .[:|||||:]
VIII
,
27
, S ={X} E, yE Xi. S E.
, S S, S S S .
, : S Xi. Xi , , , .. S
S Xi , , Xi.
27.1 (-). [a; b] - .
. : S [a; b] .
I1 = [a; b]. I1 . (.. +=).
I2. : , - I3. {In}, S. .. Ik+1 Ik, {In} (I1I2 I3 . . . ). In=
I12n1
0 {In} (...). 19.1 ...!cn=1 In. .. In [a; b], c [a; b].
S , c . (; ). = min{c ; c}. {In} 0, In, . In , , , , , In
S, , , (; ). , , , . [:|||||:]
23
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28
x E, - E.
x E. , E = (; ), E = [; ]( ,
. , (x ; x + ) ).
28.1 (-). X .
. . .. X , [a; b] :x X x [a; b]. X x [a; b]U(x), X. ? ,, y : y / [a; b]. y < a, (y; a)
X. x[a; b]U(x), .
[a; b] (.. y[a; b] : y /U(x),
U(x) U(y), ). ,
[a; b] ( 27.1). X ( ), - , , , X , . [:|||||:]
x E , E, .. E
U(x) =
. ,
, , , .
IX
29
E R a R E ( a E). f:ER. ( R) .
29.1
limExa
f(x) =b >0()> 0 :x E, 0
8/13/2019 Matan 8dec
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: . , b - a.
, , x . :
limx f(x) =b >0x0:x E,|x|> x0 |f(x) b|<
29.2
limExa
f(x) =b {xn}:n N xn=a xn E, limn
xn=a {f(xn)} n
b
: x a, f(x) b.
, , , . , .
, ? f:NR. .. Ef = N. E. , = 0.3 () E. , , N ( =a, aN, ).
. , a , lim
Exaf(x). , lim
x0
sinxx
= 1 (,
), x= 0 .
30
30.1. .
.
.
V(b)U(a) : f(U(a)E) V(b) x U(a)E f(x) V(b) .
{xn} :xn = axnn
a. -
> 0N() N n N 0 0. > 0 , >0, xnU(a) |f(xn)b|< ,..{f(xn)}
nb, .
.
25
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: limxa
f(x) = b , b= limxa
f(x) .
:
V(b)U(a) xU(a) :|f(x) b|
: 0, i
xi, |f(xi) b| 0. .. , {n}={ 1n}. , - xn, 0 0 :x U(a) |f(x) b| < . , U(a), x, b < f(x) < b+ , .. f(x). [:|||||:]
31
, >0 (a ; a) (a; a+) E.
b () f , :
>0 : ()> 0 : x E, a < x < a (a < x < a+) |f(x) b|< b () f , :
{xn}:nN, xn< a (xn> a) xn E limn
xn=a {f(xn)} n
b
: limxa0
f(x) =b = f(a0) limxa+0
f(x) =b = f(a + 0).
, ,
26
8/13/2019 Matan 8dec
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x a , , . . a , , a. :
limxa
f(x) =b f(a 0) =f(a+ 0) =b(..
xE :a
< x < a
xE :a < x < a+
xE
0 0 = 1 :x :1 < x < 0 sgn x =1 | sgn x(1)| = 0 < x : 0 < x < 1sgn x = 1 | sgn x 1| = 0 < . lim
x00f(x) =1, lim
x0+0f(x) = 1.
.. 1=1, limx0 f(x).
X
,
32 , -
. :
32.1. limxa
f(x) =b, limxa
g(x) =c. :
1. limxa
(f(x) g(x)) =b c
2. limxa
(f(x) g(x)) =bc
3. limxa
f(x)g(x)
= bc, c= 0 xU(a) :g(x)= 0
. - ( - ). , {xn} :n N xn = a : lim
nxn =
a {f(xn)} n
b, {g(xn)} n
c. , .., ,
{f(xn) g(xn)} n
b c limxa
(f(x) g(x)) =b c. [:|||||:]
, , 32.1:
( , ), .
27
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33
f(x) :X R, g(y) : Y R, x X : f(x) Y (f(X) Y). X g f :f g () g (f(x)). 33.1. f(x) :X R, g(y) :Y R, f(X) Y. lim
xx0f(x) =y0, lim
yy0g(y).
limxx0
g (f(x)) = limyy0
g(y).
. limyy0
g(y) = z0. U(z0)
V(y0) :y Y V (y0) g(y) U(z0). V(y0)W(x0) :x W(x0) f(x)Y V(y0). , :
U(z0)W(x0) :xW(x0) X f(x) Y V(y0)g (f(x))U(z0)
, . [:|||||:]
, - , , . -, , z0 - x0.
, . , , , , ,
. , limx0
(sinxx
)2. , (
) , :
f(x) = sinx
x g(x) =x2
. 33.1 f(x) g(x) f(x).
34
limxa
f(x) =b, limxa
g(x) =c b < c. :
1.U(a) :xU(a) :f(x)< g(x)
.
d: b < d < c. U1(a), U2(a) :xU1(a) |f(x) b|< 1 = d b xU2(a) |g(x) c|< 2=c d.
U(a) = U1(a) U2(a) :xU(a) |f(x) b|< d b |g(x) c|< c db d < f(x) b < d b d c < g(x) c < c df(x)< d= d c+c < g(x)
[:|||||:]
2. . f , g , h : E R x E f(x) g(x) h(x)
limxa
f(x) = limxa
h(x) =b.
limxa
g(x) =b.
28
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.
>0 U1(a) :xU1(a) |f(x) b|< U2(a) :xU2(a) |h(x) b|< b < f(x)< b+ b < h(x)< b+
xU1(a) U2(a)
b < f(x)
g(x)
h(x)< b+ |g(x) b|< limxa g(x) =b.[:|||||:]
, . , : [N, ), U(a). 34.1. lim
xaf(x) =b, lim
xag(x) =c. , U(a), :
1. f(x)< g(x)b c2. f(x) g(x)b c3. f(x)< cb c4. f(x) cb c
XI
35
, f(x) , :
>0 >0 :x, x E : 0 0x, x E U(a) |f(x) b|< 2 ,|f(x) b|< 2 .|f(x) f(x)| |f(x) b| + |b f(x)|< .
, , .
f(x), {xn}, - a ( ), xn=a.
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, {f(xn)} b. - {xn}. .. N()N :n N 0
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f(x) g(x) ( ) xa, :U(a) :xU(a) E f(x) =(x) g(x), lim
xa(x) = 1
:f xa
g.
4.
f(x) = x6
1 +x4 x0
x6, .. x6
1 +x4 =
1
1 +x4 x6
f(x) = x6
1 +x4 x
x2, .. x6
1 +x4 =
x4
1 +x4 x2
(x) , , .
36.1
1. . f xa g, g xa f.
.
f= gg = 1 fg
xaf
(.. a
1, 1a
1). [:|||||:]
2. . f xa
g, g xa
h, f xa
h.
.
U1(a) :xU1(a) E f(x) =(x) g(x)U2(a) :xU2(a) E g(x) =(x) h(x)
U(a) = U1(a) U2(a) ( , .. - a) f(x) = (x) (x) h(x)f
xah, .. lim
xa((x) (x)) = lim
xa(x) lim
xa(x) = 1. [:|||||:]
3. U(a) :f(x)= 0, g(x)= 0 xU(a) E, f xa
glimxa
f(x)g(x)
= 1. ,
limxaf(x)
g(x) , f(x) = 0, g(x) = 0 (. ).
, f(x) g(x) xa, U(a) :xU(a) E f(x) =(x) g(x), xa, lim
xa(x) = 0
f(x) = o(g(x)), x a ( ) f(x) = o(g(x)), x a.:
1
x2 = o
1
x , x x2 = o(x), x
0
1
x2 =
1
x1
x x2 =x x
31
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, 3, g(x)= 0 U(a), f= o(g)limxa
f(x)g(x)
= 0.
f(x) g(x) x a f = o(g), f(x) .
O o .
, , .
37
37.1
limx0
sin x
x = 1
. limx0+0
sinxx
limx00
sinxx
, ,
1.
A
Ox B
C
x(0; 2
). R. :
SAOB =1
2R2 sin x
SAOB () =1
2R2x
SOCB =1
2R2 tg x
,
0< sin x < x
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limx00
sin x
x =
u=xx=u
u0 + 0
x0 0
= limu0+0
sin(u)u = l imu0+0
sin u
u = 1
[:|||||:]
37.2
limx0
(1 +x)1x =e
. , .
{xk} k
0 + 0 (xk 0, xk > 0) xk < 1. nk : nk 1xk < nk + 1 1nk+1 < xk 1nk . xk 0 + 0 nk + ( , ), ( , 1):
1 + 1
nk+ 1
nk
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37.3
x0 x= 0sin xx sin x= x+ o(x)
1 cos x x22
cos x= 1 x22
+ o(x2)tg xx tg x= x+ o(x)
arcsin xx arcsin x= x+ o(x)arctg xx arctg x= x+ o(x)ex 1x ex = 1 +x+ o(x)
ln(1 +x)x ln(1 +x) =x+ o(x)(1 +x)m 1mx (1 +x)m = 1 +mx+ o(x)
:
1.
sin xxlimx0
sin x
x = 1
.
2.
(1 cos x)x2
2lim
x0
(1 cos x) 2x2
= limx0
2 2sin2x2
x2 = lim
x0
4sin2x
2
4
x
2
2= 1
3.
tg xxlimx0
tg x
x = lim
x0
sin x
x 1
cos x= 1
4.arcsin xxlim
x0
arcsin x
x = [y= arcsin x] = lim
y0
y
sin y = 1
5.
arctg xxlimx0
arctg x
x = [y= arctg x] = lim
y0
y
tg y = 1
6.
ex 1x limx0
ex 1x
= [t= ex 1] = limt0
t
ln(1 +t)= 1, ..:
limt0
ln(1 +t)t
= limt0
ln(1 +t)1t = ln lim
t0(1 +t)
1t = ln e= 1
( ) , .. - , .
7.
(1+x)m1mx limx0
(1 +x)m 1mx
= limx0
em ln(1+x) 1m ln(1 +x)
ln(1 +x) x
= limx0
em ln(1+x) 1m ln(1 +x)
ln(1 +x)x
= 1
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XII
38
f:E R. f a E,
V(f(a))U(a) :xU(a) E f(x)V(f(a)), .. f(U(a) E)V(f(a))
:
>0()> 0 :x E,|x a|< |f(x) f(a)|<
, - , , -
. , - , - a a . :
f(x) =
{x, x[0; 1]3, x= 2
, E[f] = [0;1] {2}. , a = 2 E.
U(a) :
U(a)E ={a}, f(
U(a)E) = f(a) V(f(a)) V(f(a)). , f a= 2. , , a
-E. f a,
limxa
f(x) =f(a) limxa
f(x) =f(limxa
x)
, - a. a : f(x)C(a).
f X, -.
f(x) =x R, ..a R |f(x) f(a)|=|x a|< = . f(x) = sin x R, ..aR | sin xsin a|=|2sin xa
2 cos x+a
2 | 2| sin xa
2 |
2|xa2 |=|x a|< =.
f(x) a (), limxa+0
f(x)
limxa0
f(x)
= f(a). -
, f(x) [a; b], f [a; b](.. (a; b)) a b .
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39
39.1
39.1. f(x) g(x) a, fg, fg, fg
(g= 0) a.
. (f(x) +g(x)). . lim
xaf(x) =f(a) lim
xag(x) =g(a). , -
, , limxa
(f(x) + g(x)) = f(a) + g(a), ,
(f(x) +g(x)) a. [:|||||:]
39.2
39.2. f: X Y, g : Y R, f(x) Y. f(x) C(x0), g(x) C(y0 =f(x0)), g(f(x))C(x0).
. limxx0
f(x) =f(x0) limxy0
g(x) =g(y0). , -
, , limxx0
g(f(x)) = limxy0
g(x) =g(y0) =
g(f(x0)), , g(f(x))C(x0). [:|||||:] , f(x) a, a,
.:
V(f(a)) :U(a)x(U(a) E) :f(x) /V(f(a)) -0>0 : >0xE, |x a|< , |f(x) f(a)| 0
f(x) = sgn x a= 0. limx00 =1=f(0) = 0= limx0+0 = 1. f(x) =| sgn x| a= 0, .. lim
x00f(x) = lim
x0+0f(x) = 1=f(0) = 0.
f(x) = sin 1x
a= 0, .. limx00
f(x), limx0+0
f(x), limx0
f(x).
, , -.
40 1. lim
xaf(x), f(x) a,
, a . , a , . ,
f(x) =| sgn x| f(x) ={
| sgn x|, x= 01, x= 0
, a= 0.
2. limxa0
f(x) =, limxa+0
f(x) =, =, a I . , a= 0 f(x) = sgn x.
3. limxa0
f(x), limxa+0
f(x),
a II .
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, f(x) -[a; b], f(x) , x1, x2, . . . , xn, f(x) , lim
xa+0f(x), lim
xb0f(x).
41
f(x) x1, x2 E :x1 < x2f(x1)< f(x2). f(x) x1, x2 E :x1< x2f(x1) f(x2). f(x) x1, x2 E :x1 < x2f(x1)> f(x2). f(x) x1, x2 E :x1 < x2f(x1) f(x2).
41.1. f(x) [a; b], (a; b) - .
. , f(x) [a; b]. c[a; b).
{f(x)} f(x) x(c; b]. ,.. f(b) {f(x)}. ,{f(x)} , .. f(x) f(c)x(c; b] inf{f(x)}= . , = lim
xc+0f(x). = inf{f(x)}, >0
(0; b c) :f(c + )< + . .. f(x), x(c; c + ) f(x) +. , < f(x)< + |f(x) |< lim
xc+0f(x) =.
{f(x)} f(x) x[a; c). , .. f(a) {f(x)}. ,{f(x)} , .. f(x) f(c)x[a; c) sup{f(x)} = . , = limxc0 f(x). = sup{f(x)}, > 0 (0; ca) : f(c) > . .. f(x), x (c; c) f(x) . , < f(x) < + |f(x)| < limxc0
f(x) =. [:|||||:]
41.2. f(x) . I .
. , f(x) .f(x) c(a; b), x < cf(x) f(c)f(x 0) f(c). - =, - c,
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42
42.1 ,
42.1. f:ER limxa
f(x) 0 : f(x) U(a)
E.
. limxa
f(x) =b >0()> 0 :xU(a) E |f(x) b|< b 0 :f(x) U(a).
42.2
42.3. f:ER, f C(a), aE f(a) > 0 (f(a) < 0). > 0 : f(x) >0 (f(x)< 0)
x
U(a)
E.
. f(x)C(a), >0() > 0 :xU(a) E |f(x) f(a)|0 f(a)> 0 f(a) + 0. A x[a; b] :f(x)< 0. , A= (.. aA) , ..
x
A
x < b.
sup A= p. , p
(a; b), .. f(a)< 0, f(b)> 0 42.3 1- a:x[a; a+1)f(x) < 0 2- b:x(b 2; b]f(x) > 0. , f(p) = 0. , 42.3 - p, , , .. sup A x (p;p] f(x) < 0 x (p;p+ ) f(x) 0. . [:|||||:]
43.2
43.2. fC[a; b], f(a) =, f(b) =. [min{; };max{; }]. p[a; b] :f(p) =.
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. = , = , = = , p -. = . , < < . g(x) = f(x). g(x) C[a; b] (..g(a) = f(a) = < 0, g(b) = f(b) = > 0). p : g(p) = 0, ..f(p) = 0f(p) =. [:|||||:]
43.3
43.3. f(x) [a; b] ...., .
XIV
43.4
43.4. f(x)C[a; b], f .. . f(x) . n N xn[a; b] : f(xn) > n (.. ) {f(xn)} . ..{xn} [a; b], {xn} . , - /{xnk} : lim
kxnk =p. -
p[a; b]. .. f(x)C[a; b] {f(xnk)} k
f(p).
, /
{f(xnk)
}, , -
. -. [:|||||:]
, . , f(x) = 1x
- (0; 1), .
, M (m) () f(x) E, -:
1.x E f(x) M(f(x) m).2. >0x E :f(x)> M (f(x)< m+). , , f(x) () E,
() . : M = supE
f(x), m = infE
f(x).
, . :
f(x) =
{x2, x(0; 1)12
, x {0, 1}
sup f(x) = 1 ..x[0; 1] :f(x) = 1. .
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43.5
43.5. f(x)C[a; b], (..x1, x2[a; b] :f(x1) = sup
Ef(x), f(x2) = inf
Ef(x)).
. f(x) [a; b],
. M, . . M, ..x [a; b] f(x) < M.
g(x) = 1Mf(x)
. x[a; b]Mf(x)> 0 (Mf(x))C[a; b]. - , g(x) [a; b]. -, A:x[a; b]g(x) A.Mf(x)> 0f(x) M 1
A,
, M . [:|||||:]
XV
44
44.1
f(x), (a; b). x (a; b) . x : x+ x (a; b). , x . y =f(x+x)f(x) f, .
, :
f(x)C(x)y = f(x+ x) f(x)x0
0
f x
f(x) = limx0
y
x= lim
x0
f(x+ x) f(x)x
, , f
(x0) = limx0xf(x0)f(x)
x0x . :1. f(x) =c = const.
f(x) = limx0
c cx
= 0.
2. f(x) =x.
f(x) = limx0
x+ x xx
= 1.
3. f(x) = sin x.
f(x) = limx0
sin(x+ x) sin xx
= limx0
2sinx
2 cosx+ x2
x = lim
x0cos
x+
x
2
= cos x.
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4. f(x) = cos x.
f(x) = limx0
cos(x+ x) cos xx
= limx0
2sin
x+x
2
sin
x
2
x =
= limx0
sinx+x2 = sin x.44.2
() f(x) x ( ) yx
x0 0. : f(x) f(x 0) f+(x) f(x+ 0) . :
1. f(x), f(x), f+(x), f(x) =f(x) =f+(x).
2. f
(x), f
+(x) f
(x) =f
+(x), f
(x) =f
(x) =f
+(x).3. f(x)=f+(x), f(x).
, f(x) = |x|. f(0) = limx00
|x|x
=1, f+(0) =lim
x0+0
|x|x
= 1=1 f(0).
44.3
x x+ x
M
I
P
, tg = f(x+x)f(x)x
= arctg f(x+x)f(x)x
. 0 = limIM
.
, 0 x. 0 = arctg
limx0
f(x+x)f(x)x
=
arctg f(x)tg 0 = f(x).
45
f(x) x, f(x+ x)f(x) = y =Ax+ o(x), A x, o(x) =(x)x, limx0 (x) = 0.
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45.1 ( ). f(x) x , f(x), A= f(x).
.
f(x) . x y = Ax+ o(x)
yx = A+
o(x)x = A+ o(1).
limx0
yx
=AA = f(x).
f(x) limx0
yx
= f(x). (x) = yx
f(x). limx0
(x) = 0, ..
(x) = o(1) yx
f(x) = o(1)y= f(x)x+ o(x).[:|||||:]
45.2 ( ). f(x) . x,
f(x)C(x).. y = f(x)x+ o(x)
x00 -
. [:|||||:]
, . , f(x) =|x| 0, - .
f(x) . x. y= f(x)x + o(x). , f(x)x . x dy = f(x)x. x:
x
, dx= x
.
x= (t), dx= (t)dt.
XVI
I
45.3 ( ). x = (t) . t0, y = f(x) . x0 = (t0), f((t)) . t0, f(t0) =f
(x)(t) = (f((t)))(t).
. y =f(x)x+ o(x) =f(x)x+(x)x. yt
=f(x)xt
+(x)xt
(). x = (t) , y
t t, (t).
, limt0
x = limt0
(t+ t) (t) = 0. x 0 (x) 0. () :
limt0
y
t
= limt0
f(x)x
t
=f(x)(t) =f((t))(t)
[:|||||:]
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45.4 ( ). f:XY f1 :YX . f(x)C(x0), f1(y)C(y0 =f(x0)). f(x). x0 f
(x0)= 0, f1(y) . y0, (f1)(y0) = 1f(x0) .. x x0 X y y0 Y. , f1(y0) = x0, f1(y0 + y) =x0+ x, .. x= f
1(y0+ y)
f1(y0).
(f1)(y0) = limy0
f1(y0+ y) f1(y0)y
= limx0
x0+ x x0f(x0+ x) f(x0)=
= limx0
1
f(x0+ x) f(x0)x
= 1
f(x0)
[:|||||:]
45.5( ). f(x) g(x) . x, f
g, f
g, f
g (g
= 0) . x, (f
g) =
f g, (f g) =fg+f g,fg
= fgfgg2
.
. , f f(x),g g(x), h h(x).
1. h(x) =f(x) g(x).
h= h(x+ x) h(x) = (f(x+ x) g(x+ x)) (f(x) g(x)) == (f(x+ x) f(x)) (g(x+ x) g(x)) = f g
,h
x=
f
x g
x
x0 , f(x) g(x),
h(x) =f(x) g(x).2. h(x) =f(x) g(x).
h= h(x+ x) h(x) =f(x+ x)g(x+ x) f(x)g(x) == (f(x+ x)g(x+ x)
f(x+ x)g(x)) + (f(x+ x)g(x)
f(x)g(x))
h= f(x+ x)(g(x+ x) g(x)) +g(x)(f(x+ x) f(x)) =f(x+ x)g+g(x)f
,h
x=f(x+ x)
g
x+g(x)
f
x
x 0. f(x) x (.. . ) lim
x0
f(x+ x) =f(x). ,
h(x) =f(x)g(x) +g(x)f(x).
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3. h(x) = f(x)g(x)
. g(x)= 0, g(x+ x)= 0 x.
h= h(x+ x) h(x) = f(x+ x)g(x+ x)
f(x)g(x)
=f(x+ x)g(x) g(x+ x)f(x)
g(x)g(x+ x) =
=
(f(x+ x)g(x)
f(x)g(x))
(g(x+ x)f(x)
f(x)g(x))
g(x)g(x+ x) =
=g(x)(f(x+ x) f(x)) f(x)(g(x+ x) g(x))
g(x)g(x+ x) =
g(x)f f(x)gg(x)g(x+ x)
h
x=
g(x)f
x f(x) g
xg(x)g(x+ x)
, ,
h(x) =g(x)f(x) f(x)g(x)
g2(x) .
[:|||||:]
45.1 ,
y x t, ..
{x= (t)
y= (t),
tT, , y=y(x) , t .
45.6 ( , ). . T, x0 , , ..t= 1(x). y(x) =
(t)(t)
.
. , y = (t) = (1(t)). : dx = (t)dt, dy = (t)dt.
dy=y (x)dx, dydx
= (t)
(t). [:|||||:]
XVII
II
, y=f(x) () c, U(c) :xU(c) x f(c)), x > cf(x) > f(c) (f(x) < f(c)). y =f(x) () c, U(c) :x U(c) f(x) < f(c) (f(x) >f(c)).
, ( ),
/ . y =f(x) , - .
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45.7 ( / ). y = f(x). c f(c)> 0 (f(c)< 0), f(x) () c.
. f(c)> 0. .
f(c) = limxc
f(x) f(c)x c
>0
()> 0 :
x, 0 m, (a, b)
. f() = 0.
[:|||||:]
45
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.. , 1 b a. , . :
f(x) = {x, 0 x 0 (f(x)< 0), f(x)
( ) (a, b).
. [x1, x2](a, b). - , f(x2)f(x1)
x2x1=f().
.. x2 > x1 f()> 0, f(x2)> f(x1). , f
()< 0, f(x2)< f(x1). [:|||||:]
45.12 ( ). , - f(x) = const [a, b] , f(x) = 0x[a; b].
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. . -. [x1, x2][a, b]. ,
f(x2) f(x1)x2 x1 =f
() = 0f(x2) =f(x1) (.. x2=x1) f(x) =const
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45.13 ( ). x(t) y(t):
1. x(t), y(t)C[, ].2. x(t), y(t) . (, ).
:
1.
(, ) :x()(y()
y()) =y ()(x()
x()).
2. x(t)= 0 t[, ], y()y()x()x()
= y()x()
.
.
1. F(t) =x(t)(y() y()) y(t)(x() x()). - 1 2. ,
F() =x()y() x()y() y()x() +y()x() =x()y() y()x()F() =x()y() x()y() y()x() +y()x() =x()y() +y()x()
..F() =F()
, (, ) :F
() = 0.
F
() =x
()(y() y()) y()(x() x()), .. x()(y() y()) =y ()(x() x()).2. x()= 0, x()=x(), .. .
1 .
[:|||||:]
, , . y(t) =t, . :
x() x()y() y()
=x()
y
()x() x()
=x()
x()
x() =x()(
)
x() =x(), .
x() x() =x()( ) = 0x() = 0
XIX
, f(x) E,
>0()> 0 :x, x E,|x x|< |f(x) f(x)|<
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x0:
>0(, x0)> 0 :x E,|x x0|< |f(x) f(x0)|< , -
, . , , , x. -
x0, x .
45.14. f(x) E, f(x) E.
. x =x0, x =x, -
x0, x0 E. [:|||||:] 45.15. f(x) E, f(x) E E.
. . [:|||||:]
45.16. f(x) E, , , - E.
. -. f(x) = x2 E = (1, +). - :
0>0 : >0x, x (1, +), |x x|< |f(x) f(x)| 0 , x1 =x
, x2 = x.|f(x) f(x)|=|x21 x22|=|x1 x2||x1+ x2|.
, |x1 x2|= 2 . , |x1+ x2| . ,x1, x2(1, +) :|x1+x2|> 3 . , |x1 x2||x1+x2|> 32 =0. [:|||||:]
f(x2)
f(x2)
f(x2) +
f(x1)
f(x1) +
f(x1)
(x1 , x1 + ) (x2 , x2 + )
. f(x) = x2. , - , x - f(x) . f(x) , , - x - . , .
, , - .
45.17 ( ). f(x)C[a, b], f(x) [a, b].
. . 0 > 0 : >0x, x [a, b], |x x|< |f(x) f(x)| 0. = 1n , nN. nNxn, xn [a, b] :|xn xn| < 1n |f(xn) f(xn)| 0. .. xn, xn [a, b], {xn} {xn} , {xkn} {xn} {xkn} {xn}, lim
nxkn = , limn
xkn = . ..n N |xkn
xkn|< 1
n, =. xkn x
kn
[a, b], f(x) . , limn
f(xkn) =
f(), limn
f(xkn) =f() =f(). .. lim
nf(xkn) = limn
f(xkn). ,
|f(xn) f(xn)| 0. [:|||||:]
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XX
46
f(x), -E. ..f(x). f(x) . E, (f(x)) = f(x). , f(x) . E, f(x) = f(3)(x). n- f(n)(x) = (f(n1)(x)), f(0)(x) = f(x). -, n- .
, n E C(n)(E).
:
f(x) = sin x.
f(x) = cos x= sin
2+x
f(x) = sin x= sin (+x)f(x) = cos x= sin
3
2 +x
f(4)(x) = sin x
, f(n)(x) = sin
(n2
+x
). n = 1 .
n, n= n+ 1.
f(n+1)(x) = sin
(n+ 1)
2 +x
f(n+1)(x) = (f(n)(x)) =
sinn
2 +x
= cos
n2
+x
= sin
(n+ 1)
2 +x
f(x) =ex. f(n)(x) =ex.
f(x) =xm. n , f(n)(x) =m(m 1) . . . (m n + 1)xmn.
f(x) = ln x. f
(x) = 1x =x
1
. f(n)
(x1
) = (1)(2) . . . (n)x1n
= (1)n
n! x1n
. , f(n)(x) =f(n1)(x1) = (1)n1(n 1)!xn.
47
47.1. u(x) v(x) n E.
(u v)(n) =n
k=0
Ckn u(nk) v(k).
. . n = 1, (u
v) = uv+ uv =1
k=0 Ck1 u(nk) v(k). n, n = n+ 1.
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(u v)(n+1):
(u v)(n+1) =
nk=0
Ckn u(nk) v(k)
=n
k=0
Ckn u(nk+1) v(k) +n
k=0
kn u(nk) v(k+1) =
=
nk=0
Ckn u
(nk+1)
v(k)
+
n+1k=1
Ck1n u
(nk+1)
v(k)
=C0n u
(n+1)
v +n
k=1
(Ckn+Ck1n ) u(nk+1) v(k)++ Cnn u v(n+1) =C0n u(n+1) v +
nk=1
Ckn+1 u(nk+1) v(k) + Cnnu v(n+1) =n+1k=0
Ckn+1 u(nk+1) v(k)
:
Ckn+Ck1n =
n!
k!(n k)!+ n!
(k 1)!(n k+ 1)! =(n k+ 1)n! +kn!
k!(n k+ 1)! = (n+ 1)!
k!(n k+ 1)! =Ckn+1
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