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    Теория множеств

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.,

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n

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2. '

n=1

.

Sk =1+2+3+4+…+k=

k (k + 1)
.'
2

,

Sk+1 =

(k + 1)(k + 2)
.
2

14
,
Sk+1 = Sk +(k+1)=
4

*

.

4. 1

'

k (k + 1)
(k + 1)(k + 2)
+ (k+1)=
.
2
2

,

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9.

#

. +

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13+23+33

A"

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,

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k3+(k+1)3+(k+2)3,

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9. 4

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,
# 1.

# #

k-

,

9.

#

(k+1)3+(k+2)3 +(k+3)3=(k+1)3+(k+2)3+ k3+9k2+27k+27=[k3+(k+1)3+(k+2)3]+9(k2+3k+3)
#
#

",

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#

9,

9.

4&1&
4

1. ,
Sn= 1+2+ 22 +23 +… 2n-1.

4

2. 1

,

n

#

1⋅2+2⋅5+…+n(3n-1)= n2 (n+1) .
4

3. 1

,

n

#

12 +22 +32 +…+n2 =
4

4. 1

,

n(n + 1)(2n + 1)
6
n

#

n(4n 2 − 1)
1 +3 +5 +…+(2n-1) =
.
3
2

4

5. 1

,

2

2

2

n

#

1⋅2+2⋅3+3⋅4+…+(n-1)n=
4

6. 1

,

(n − 1)n(n + 1)
.
3
n

#

13 +23 +33 +…+n3 = (

n(n + 1) 2
).
2

4

7. 1

,

n

5⋅23n-2 + 33n-1

4

8. 1

,

n

n(2n2-3n+1)

4

9. 1

,

n

n5 –n

4

10. 1

,

#

n

#
#

#

5.

19.
6.

15
1
1
1
n
+
+ …+
=
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(2n − 1)(2n + 1) 2n + 1
11. 1

4

,

n

#

1
1
1
1
n
+
+
+ …+
=
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1 ⋅ 4 4 ⋅ 7 7 ⋅ 10
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4

12. '

4

13. 1

2n > n2 ?

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"

: ∀α > −1

∀n ∈ N : (1 + α ) n ≥ 1 + nα (

).

6

1
1
1
1
+ 2 + ⋅⋅⋅ + 2 ≤ 2 − .
2
n
2
3
n

4

14. 1

,

∀n ∈ N : 1 +

4

15. 1

,

∀n ∈ N : 2 n > n.

,
2004-2005

1
4

1. 7

4

2. .

4

3. -

4

4. '

".

"

inf

(

.

inf
E=

.

Eα ,

9 ∈A

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(

supy9=y0.

(

α∈A

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4

5. ,

4

6. $ *

4

7. ,

4

8. '

4

9. '

,

y0= supE.

&: , &\ , \&, &

&={-1;0;1;2;3}, ={2;3;4;5}.

: |x|>|x+1|.
inf X

sup X,

X={xn: xn=3 sin 4n, n ∈ N}.
,

?

,

?

(" ∈ ?). 1
4

,

inf X=0

sup X=3.

, -? –

,

–"

5

inf (–X)= - sup X.

10. 1

,

(

n2-6n+12>0

#

n ∈ N.
2
4

1. 7

4

2. .

4

3. -

4

4. '
n ∈ N,

".

"

sup

(

sup
,

.
!

,

.

"

"=0.

,

# "B0

n">

#

#

"

16
4

5. ,

4

6. $ *

4

7. ,

4

8. '

4

9. '

&: , &\ , \&, &

&={-1;0;1;2;3;4;5;6}, ={2;3;4;5}.

: |x+2|+|x-1|>12.
inf X

sup X,

X={xn: xn=0,5 cos 7n, n∈N}.
,

?

,

?

(" ∈ ?). 1
4

,

inf X=-1

sup X=2.

, -? –

,

–"

5

sup( –X)= - inf X.

10. 1

6n-n2-12<0

,

(

#

n∈N.
3
4

1. 7

4

2. .

4

3. -

4

4. 1

".

"

inf

(

,
,

X\(

5. ,

4

6. $ *

4

7. ,

4

8. '

4

9. '

" .

A α )=

α ∈!

4

α ∈!

&: , &\ , \&, &

(X\ A α ).

,

&={-1;0;1;2;3}, ={-3;-2;-1;0;1;2;3;4;5}.

: |2"-1|<|x-1|.
inf X

sup X,

X={xn: xn=3 sin
,

?
?,Y

3
, n ∈ N}.
n
inf X=-2

,

10. 1

sup X=2.

,Z–

z=x+y (" ∈ ?, y ∈ Y). 1
4

.

,

5

inf Z= inf X + inf Y.
,

(

10-3n- n2>0

nB2, n ∈ N.
4
4

1. 7

4

2. .

4

3. -

4

4. 1

sup

(

,

X\(

5. ,

4

6. $ *

4

7. ,

4

8. '

4

9. '

A α )=

.

"

,
α ∈!

4

".

"

" .
α ∈!

&: , &\ , \&, &

(X\A α ).
,

&={-1;0;1;2;3}, ={-3;-2;-1;0}.

: |"+2|-|x-3|>-5.
inf X

sup X,

X={xn: xn= - 2 sin
?

?,Y

z=x+y (" ∈ ?, y∈Y). 1

,

,

1
, n∈N}.
n

inf X=0

sup X=10.

,Z–
,

sup Z= sup ? + sup Y.

5

#

17
4

10. 1

n2-3n+7B0

,

(

#

n∈N.
5
4

1. 7

4

2. .

4

3. -

4

4. 1

4

5. ,

4

6. $ *

4

7. ,

4

8. '

4

9. '

".

"

inf

(

.
,

(&\ ):( \&)=Ø.

&: , &\ , \&, &

,

&={-1;0;1;2;3}, ={0;10;20}.

: |"2+4|+|x-2|<8.
inf X

X={xn: xn=5 + (-1)n

sup X,

,

?
?,Y

1
, n∈N}.
n

inf X=2

sup X=3.

,
z=x⋅y (" ∈ ?, y∈Y). 1

10 1

, Z –

"

5
4

.

(

,

supZ=sup? ⋅ supY.

,

2n-n2<0

#

n>2, n ∈ N.
6
4

1. 7

4

2. .

4

3. -

4

4. '

".

"

sup

(

,
E=

.

"

.

Eα ,

9 ∈A

inf E9=y9

(

inf y9=y0.

(

α ∈A

'
4

5. ,

4

6. $ *

4

7. ,

4

8. '

4

9. '

,

y0= inf E.

&: , &\ , \&, &

&=(-5; 3), =[-2; 4].

: |"2-4|+|x-2|<8.
inf X

2
X={xn: xn=-4 + (-1)n , n ∈ N}.
n

sup X,

?
?,Y

10. 1

inf X=3

sup X=+C.
"

z=x⋅y (" ∈ ?, y ∈ Y). 1
(

n∈N.

,

,

5
4

,

,

, Z –

inf Z= inf ? ⋅ inf Y.
,

4n2-12n+9B0

#