Числовые и функциональные ряды. Учебник

4) 14;

5) 16.

Ɂɚɞɚɧɢɟ 12

ɂɫɩɨɥɶɡɭɹ ɩɪɟɞɟɥɶɧɵɣ ɩɪɢɡɧɚɤ ɞɥɹ ɢɫɫɥɟɞɨɜɚɧɢɹ ɧɚ ɫɯɨɞɢɦɨɫɬɶ ɪɹɞɚ

∞+

1

Ⱥ:

1)

2)

3)

4)

¦

n

¦

n

¦

n

¦

2

=

2

∞+

;

n

=

11n

∞+

1

;

2

n

=

1

∞+

1

−

=

2

∞+

n

=

1lnn

, ɪɹɞ ȼ ɦɨɠɟɬ ɩɪɢɧɢɦɚɬɶ ɜɢɞ:

++

2

nn

;

32

nn

;

5) ɫɪɟɞɢ ɩɪɢɜɟɞɟɧɧɵɯ ɧɟɬ ɜɟɪɧɨɝɨ ɨɬɜɟɬɚ.

Ɂɚɞɚɧɢɟ 13

ɋɭɦɦɚ ɪɹɞɚ

1

;

1)

−

5

1

;

2)

5

1

;

3)

−

3

1

;

4)

3

1

.

5)

2

¦

n

∞+

=

1

n

()

−

1

4

1

ɪɚɜɧɚ:

n

Ɂɚɞɚɧɢɟ 14

ɋɭɦɦɚ ɪɹɞɚ

5

;

1)

7 6

;

2)

7

¦

n

∞+

§ ¨ ¨ ©

=

1

10

+

1

nn

·

38

−

¸

ɪɚɜɧɚ:

n

¸ ¹

61

4

3)

;

7

1

;

4)

2

5)

.

9

Ɂɚɞɚɧɢɟ 15

ɋɭɦɦɚ ɪɹɞɚ

3

;

1)

5

4

;

2)

5

2

;

3)

5

9

1

;

4)

10

7

.

5)

10

Ɂɚɞɚɧɢɟ 16

ɋɭɦɦɚ ɪɹɞɚ

1) 1;

1

;

2)

2

3) 2;

1

;

4)

3

.

5)

2

Ɂɚɞɚɧɢɟ 17

ɋɭɦɦɚ ɪɹɞɚ

0− ;

1)

2)

3)

4)

5)

;

17,0−

;

064,0−

;

42,0−

.

032,0−

∞+

¦

=

1

n

1

⋅ nn

∞+

¦

=

2

n

§ ¨

2

©

+

⋅

n

−

()

1

−

6

· ¸

−+

5129

nn

¹

...

75153131

⋅

1

ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ 0,01 ɩɪɢɛɥɢɡɢɬɟɥɶɧɨ ɪɚɜɧɚ:

n

5

ɪɚɜɧɚ:

1

++

()

+

2

ɪɚɜɧɚ:

62

Ɂɚɞɚɧɢɟ 18

∞+

ɋɭɦɦɚ ɪɹɞɚ

()

¦

=

1

n

nnn

+++−+

1223

ɪɚɜɧɚ:

1) 1;

2) 2;

3) 3;

4) 4;

5) 5.

Ɂɚɞɚɧɢɟ 19

Ⱦɥɹ ɪɹɞɨɜ

Ⱥ:

ȼ:

ɋ:

∞+

¦

=

0

n

∞+

()

¦

=

n

∞+

§ ¨

¦

©

=

12n

−02

)2/(

n

,

5ln

n

,

·

ɜɵɛɪɚɬɶ ɜɟɪɧɨɟ ɭɬɜɟɪɠɞɟɧɢɟ:

−+

nnn

¸ ¹

1) ɜɫɟ ɫɯɨɞɹɬɫɹ;

2) ɜɫɟ ɪɚɫɯɨɞɹɬɫɹ;

3) A ɢ B – ɪɚɫɯɨɞɹɬɫɹ, ɋ – ɫɯɨɞɢɬɫɹ;

4) A ɢ ɋ – ɫɯɨɞɹɬɫɹ, B – ɪɚɫɯɨɞɢɬɫɹ;

5) ȼ ɢ ɋ – ɪɚɫɯɨɞɹɬɫɹ ɭɫɥɨɜɧɨ, Ⱥ – ɫɯɨɞɢɬɫɹ.

Ɂɚɞɚɧɢɟ 20

Ⱦɥɹ ɪɹɞɨɜ

Ⱥ:

ȼ:

¦

n

¦

n

∞+

2

n

=

1

§

1

+

¨

∞+

©

e

=

1

sin

2

n

1

· ¸

n

¹

n

,

1

++

nn

2

ɜɵɛɪɚɬɶ ɜɟɪɧɨɟ ɭɬɜɟɪɠɞɟɧɢɟ:

1) Ⱥ ɢ ȼ – ɧɟɨɛɯɨɞɢɦɨɟ ɭɫɥɨɜɢɟ ɜɵɩɨɥɧɟɧɨ, ɪɹɞ ɫɯɨɞɢɬɫɹ;

2) Ⱥ ɢ ȼ – ɧɟɨɛɯɨɞɢɦɨɟ ɭɫɥɨɜɢɟ ɧɟ ɜɵɩɨɥɧɟɧɨ, ɪɹɞ ɪɚɫɯɨɞɢɬɫɹ;

3) A – ɧɟɨɛɯɨɞɢɦɨɟ ɭɫɥɨɜɢɟ ɜɵɩɨɥɧɟɧɨ, ɪɹɞ ɪɚɫɯɨɞɢɬɫɹ, ȼ – ɧɟɨɛɯɨɞɢɦɨɟ

ɭɫɥɨɜɢɟ ɜɵɩɨɥɧɟɧɨ, ɪɹɞ ɫɯɨɞɢɬɫɹ;

4) A – ɧɟɨɛɯɨɞɢɦɨɟ ɭɫɥɨɜɢɟ ɜɵɩɨɥɧɟɧɨ, ɪɹɞ ɫɯɨɞɢɬɫɹ, B – ɧɟɨɛɯɨɞɢɦɨɟ

ɭɫɥɨɜɢɟ ɜɵɩɨɥɧɟɧɨ, ɪɹɞ ɪɚɫɯɨɞɢɬɫɹ;

5) ɨɬɜɟɬ ɨɬɥɢɱɟɧ ɨɬ ɩɪɢɜɟɞɟɧɧɵɯ.

63

Ɂɚɞɚɧɢɟ 21

ɉɪɢɦɟɧɹɹ ɫɜɨɣɫɬɜɚ ɥɢɧɟɣɧɵɯ ɨɩɟɪɚɰɢɣ ɧɚɞ ɱɢɫɥɨɜɵɦɢ ɪɹɞɚɦɢ, ɢɫɫɥɟɞɭɣɬɟ

ɫɯɨɞɢɦɨɫɬɶ ɪɹɞɨɜ

Ⱥ:

ȼ:

∞+

¦

=

1

n

∞+

n

¦

()

nn

=

1

n

+

2

13

nnn

1175

++

,

n

:

1

1) ɨɛɚ ɪɚɫɯɨɞɹɬɫɹ;

2) ɨɛɚ ɫɯɨɞɹɬɫɹ;

3) A – ɫɯɨɞɢɬɫɹ, ȼ – ɪɚɫɯɨɞɢɬɫɹ;

4) A – ɪɚɫɯɨɞɢɬɫɹ, B – ɫɯɨɞɢɬɫɹ;

5) ɨɬɜɟɬ ɨɬɥɢɱɟɧ ɨɬ ɩɪɢɜɟɞɟɧɧɵɯ.

Ɂɚɞɚɧɢɟ 22

ɂɫɫɥɟɞɭɣɬɟ ɫɯɨɞɢɦɨɫɬɶ ɪɹɞɨɜ, ɩɪɢɦɟɧɹɹ ɩɪɢɡɧɚɤ ɫɪɚɜɧɟɧɢɹ

∞+

1

Ⱥ:

ȼ:

¦

n

¦

n

=

1

∞+

n

2

=

1

,

1

+

nn

2

:

2

n

1) Ⱥ ɢ ȼ – ɫɯɨɞɹɬɫɹ;

2) Ⱥ ɢ ȼ – ɪɚɫɯɨɞɹɬɫɹ;

3) A – ɫɯɨɞɢɬɫɹ, ȼ – ɪɚɫɯɨɞɢɬɫɹ;

4) A – ɪɚɫɯɨɞɢɬɫɹ, B – ɫɯɨɞɢɬɫɹ;

5) ɩɪɢɡɧɚɤ ɫɪɚɜɧɟɧɢɹ ɩɪɢɦɟɧɢɬɶ ɧɟɥɶɡɹ.

Ɂɚɞɚɧɢɟ 23

ɉɪɢɦɟɧɹɹ ɩɪɢɡɧɚɤ Ⱦɚɥɚɦɛɟɪɚ, ɢɫɫɥɟɞɭɣɬɟ ɫɯɨɞɢɦɨɫɬɶ ɪɹɞɨɜ

Ⱥ:

ȼ:

¦

n

¦

n

∞+

=

1

∞+

1

=

2

nn

+

2

n

,

n

+

3

π

n

sin

n

2

:

!

n

1) Ⱥ ɢ ȼ – ɫɯɨɞɹɬɫɹ;

2) Ⱥ ɢ ȼ – ɪɚɫɯɨɞɹɬɫɹ;

3) A – ɫɯɨɞɢɬɫɹ, ȼ – ɪɚɫɯɨɞɢɬɫɹ;

4) A – ɪɚɫɯɨɞɢɬɫɹ, B – ɫɯɨɞɢɬɫɹ;

5) ɩɪɢɡɧɚɤ Ⱦɚɥɚɦɛɟɪɚ ɩɪɢɦɟɧɢɬɶ ɧɟɥɶɡɹ.

64

Ɂɚɞɚɧɢɟ 24

ɉɪɢɦɟɧɹɹ ɪɚɞɢɤɚɥɶɧɵɣ ɩɪɢɡɧɚɤ Ʉɨɲɢ, ɢɫɫɥɟɞɭɣɬɟ ɫɯɨɞɢɦɨɫɬɶ ɪɹɞɨɜ

Ⱥ:

ȼ:

¦

n

¦

n

∞+

=

∞+

=

n

π

,

2

1

n

+

1

·

§ ¸

¨

n

¹

©

n

2

:

n

2

⋅

nn

1) Ⱥ ɢ ȼ – ɫɯɨɞɹɬɫɹ;

2) Ⱥ ɢ ȼ – ɪɚɫɯɨɞɹɬɫɹ;

3) A – ɫɯɨɞɢɬɫɹ, ȼ – ɪɚɫɯɨɞɢɬɫɹ;

4) A – ɪɚɫɯɨɞɢɬɫɹ, B – ɫɯɨɞɢɬɫɹ;

5) ɪɚɞɢɤɚɥɶɧɵɣ ɩɪɢɡɧɚɤ Ʉɨɲɢ ɩɪɢɦɟɧɢɬɶ ɧɟɥɶɡɹ.

Ɂɚɞɚɧɢɟ 25

ɉɪɢɦɟɧɹɹ ɢɧɬɟɝɪɚɥɶɧɵɣ ɩɪɢɡɧɚɤ Ʉɨɲɢ, ɢɫɫɥɟɞɭɣɬɟ ɫɯɨɞɢɦɨɫɬɶ ɪɹɞɨɜ

∞+

1

Ⱥ:

ȼ:

¦

n32

¦

n

=

∞+

=

2

,

nn

ln

n

+

111

§

ln

¨ ©

·

:

−

¸

−

nn

¹

1) Ⱥ ɢ ȼ – ɫɯɨɞɹɬɫɹ;

2) Ⱥ ɢ ȼ – ɪɚɫɯɨɞɹɬɫɹ;

3) A – ɫɯɨɞɢɬɫɹ, ȼ – ɪɚɫɯɨɞɢɬɫɹ;

4) A – ɪɚɫɯɨɞɢɬɫɹ, B – ɫɯɨɞɢɬɫɹ;

5) ɢɧɬɟɝɪɚɥɶɧɵɣ ɩɪɢɡɧɚɤ Ʉɨɲɢ ɩɪɢɦɟɧɢɬɶ ɧɟɥɶɡɹ.

Ɂɚɞɚɧɢɟ 26

ɋɤɨɥɶɤɨ ɱɥɟɧɨɜ ɪɹɞɚ

ɫɭɦɦɭ ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ

1) 894;

2) 889;

3) 900;

4) 999;

5) 10000.

∞+

¦

1

=

n

310−

?

1

−

n

()

1

−

2

1

+

n

ɧɟɨɛɯɨɞɢɦɨ ɜɡɹɬɶ, ɱɬɨɛɵ ɜɵɱɢɫɥɢɬɶ ɟɝɨ

65

Ɂɚɞɚɧɢɟ 27

ɉɪɢɦɟɧɹɹ ɩɪɢɡɧɚɤ Ʌɟɣɛɧɢɰɚ, ɢɫɫɥɟɞɭɣɬɟ ɫɯɨɞɢɦɨɫɬɶ ɪɹɞɨɜ

Ⱥ:

ȼ:

¦

n

¦

n

∞+

= ∞+

=

1

()

1

−

()

1

−

n

1

n

11−

,

tg

n

1

−

n

:

20

+

1) Ⱥ ɢ ȼ – ɫɯɨɞɹɬɫɹ;

2) Ⱥ ɢ ȼ – ɪɚɫɯɨɞɹɬɫɹ;

3) A – ɫɯɨɞɢɬɫɹ, ȼ – ɪɚɫɯɨɞɢɬɫɹ;

4) A – ɪɚɫɯɨɞɢɬɫɹ, B – ɫɯɨɞɢɬɫɹ;

5) ɩɪɢɡɧɚɤ Ʌɟɣɛɧɢɰɚ ɩɪɢɦɟɧɢɬɶ ɧɟɥɶɡɹ.

Ɂɚɞɚɧɢɟ 28

Ⱦɥɹ ɪɹɞɨɜ

n

·

()

−⋅+

154

¸

,

¸ ¹

1

ɜɵɛɪɚɬɶ ɜɟɪɧɨɟ ɭɬɜɟɪɠɞɟɧɢɟ:

Ⱥ:

ȼ:

¦

n

¦

n

∞+

1

=

∞+

=

1

§ ¨ ¨

10

©

−

n

()

1

−

n

1) Ⱥ ɢ ȼ – ɪɚɫɯɨɞɹɬɫɹ;

2) Ⱥ ɢ ȼ – ɫɯɨɞɹɬɫɹ ɚɛɫɨɥɸɬɧɨ;

3) A – ɫɯɨɞɢɬɫɹ ɚɛɫɨɥɸɬɧɨ, ȼ – ɫɯɨɞɢɬɫɹ ɭɫɥɨɜɧɨ;

4) Ⱥ – ɫɯɨɞɢɬɫɹ ɚɛɫɨɥɸɬɧɨ, ȼ – ɪɚɫɯɨɞɢɬɫɹ;

5) Ⱥ – ɫɯɨɞɢɬɫɹ ɭɫɥɨɜɧɨ, ȼ – ɫɯɨɞɢɬɫɹ ɚɛɫɨɥɸɬɧɨ.

Ɂɚɞɚɧɢɟ 29

ɉɪɢɦɟɧɹɹ ɩɪɟɞɟɥɶɧɵɣ ɩɪɢɡɧɚɤ, ɢɫɫɥɟɞɭɣɬɟ ɫɯɨɞɢɦɨɫɬɶ ɪɹɞɨɜ

∞+

Ⱥ:

¦

=

n

1

∞+

ȼ:

¦

=

1

n

1) Ⱥ ɢ ȼ – ɫɯɨɞɹɬɫɹ;

2) Ⱥ ɢ ȼ – ɪɚɫɯɨɞɹɬɫɹ;

3) A – ɫɯɨɞɢɬɫɹ, ȼ – ɪɚɫɯɨɞɢɬɫɹ;

4) A – ɪɚɫɯɨɞɢɬɫɹ, B – ɫɯɨɞɢɬɫɹ;

5) ɩɪɟɞɟɥɶɧɵɣ ɩɪɢɡɧɚɤ ɩɪɢɦɟɧɢɬɶ ɧɟɥɶɡɹ.

π

n

,

sin2

n

3

()

1

−

n

24

++

nn

:

23

66

Ɂɚɞɚɧɢɟ 30

ȼɨɫɶɦɨɣ ɷɥɟɦɟɧɬ ɱɢɫɥɨɜɨɝɨ ɪɹɞɚ 1;

31

;

1)

3

64

;

2)

12,4

33

;

3)

3

64

17

4)

;

4

81 17

5)

.

4

64

1

7

;

2

4

1

;

; … ɪɚɜɟɧ:

3

9

16

67

Ɋɚɡɞɟɥ II

x

ɎɍɇɄɐɂɈɇȺɅɖɇɕȿ ɊəȾɕ

Ɉɬ ɪɚɫɫɦɨɬɪɟɧɧɵɯ ɜ ɩɟɪɜɨɦ ɪɚɡɞɟɥɟ ɱɢɫɥɨɜɵɯ ɪɹɞɨɜ ɩɟɪɟɣɞɟɦ ɤ ɪɹɞɚɦ ɮɭɧɤɰɢɨɧɚɥɶɧɵɦ, ɢɦɟɸɳɢɦ ɨɝɪɨɦɧɨɟ ɩɪɚɤɬɢɱɟɫɤɨɟ ɡɧɚɱɟɧɢɟ ɩɪɢ ɪɟɲɟɧɢɢ ɫɚɦɵɯ ɪɚɡɧɨɨɛɪɚɡɧɵɯ ɡɚɞɚɱ. Ɉɫɨɛɨɟ ɜɧɢɦɚɧɢɟ ɜ ɞɚɧɧɨɦ ɪɚɡɞɟɥɟ ɛɭɞɟɬ ɭɞɟɥɟɧɨ ɱɚɫɬɧɵɦ ɫɥɭɱɚɹɦ ɮɭɧɤɰɢɨɧɚɥɶɧɵɯ ɪɹɞɨɜ – ɫɬɟɩɟɧɧɵɦ ɪɹɞɚɦ ɢ ɪɹɞɚɦ Ɏɭɪɶɟ.

Ƚɥɚɜɚ 4

ɎɍɇɄɐɂɈɇȺɅɖɇɕȿ ɊəȾɕ:

ɈȻɓɂȿ ɈɉɊȿȾȿɅȿɇɂə ɂ ɋȼɈɃɋɌȼȺ

ȼ ɞɚɧɧɨɣ ɝɥɚɜɟ ɨɫɬɚɧɨɜɢɦɫɹ ɧɚ ɨɛɳɢɯ ɞɥɹ ɮɭɧɤɰɢɨɧɚɥɶɧɵɯ ɪɹɞɨɜ ɩɨɧɹɬɢɹɯ, ɨɩɪɟɞɟɥɟɧɢɢ ɨɛɥɚɫɬɢ ɫɯɨɞɢɦɨɫɬɢ. Ɋɚɫɫɦɨɬɪɢɦ ɩɪɚɜɢɥɶɧɨ ɫɯɨɞɹɳɢɟɫɹ ɪɹɞɵ ɢ ɢɯ ɫɜɨɣɫɬɜɚ.

4.1. Ɉɛɥɚɫɬɶ ɫɯɨɞɢɦɨɫɬɢ ɮɭɧɤɰɢɨɧɚɥɶɧɨɝɨ ɪɹɞɚ

Ɋɹɞɵ ɜɢɞɚ

() () () ()

21

ɱɥɟɧɚɦɢ ɤɨɬɨɪɵɯ ɹɜɥɹɸɬɫɹ ɮɭɧɤɰɢɢ ɤɨɬɨɪɨɣ ɨɛɥɚɫɬɢ ɢɡɦɟɧɟɧɢɹ ɚɪɝɭɦɟɧɬɚ

ȿɫɥɢ

ɤɚɤɨɟ-ɥɢɛɨ ɱɢɫɥɨɜɨɟ ɡɧɚɱɟɧɢɟ

, ɬɨ ɢɦɟɟɦ ɱɢɫɥɨɜɨɣ ɪɹɞ

()

xu

n

() ()

, ɧɚɡɵɜɚɸɬɫɹ ɮɭɧɤɰɢɨɧɚɥɶɧɵɦɢ.

() () ()

ɤɨɬɨɪɵɣ ɦɨɠɟɬ ɫɯɨɞɢɬɶɫɹ ɢɥɢ ɪɚɫɯɨɞɢɬɶɫɹ. ȿɫɥɢ ɪɹɞ (4.2) ɫɯɨɞɢɬɫɹ, ɬɨ ɬɨɱɤɚ ɧɚɡɵɜɚɟɬɫɹ ɪɹɞ (4.2) ɪɚɫɯɨɞɢɬɫɹ, ɬɨ ɬɨɱɤɚ

ɬɨɱɤɨɣ ɫɯɨɞɢɦɨɫɬɢ ɮɭɧɤɰɢɨɧɚɥɶɧɨɝɨ ɪɹɞɚ (4.1). ȿɫɥɢ ɩɪɢ

ɧɚɡɵɜɚɟɬɫɹ ɬɨɱɤɨɣ ɪɚɫɯɨɞɢɦɨɫɬɢ ɮɭɧɤɰɢɨ-

x

0

ɧɚɥɶɧɨɝɨ ɪɹɞɚ (4.1). Ⱦɥɹ ɨɞɧɢɯ ɬɨɱɟɤ, ɜɡɹɬɵɯ ɢɡ ɨɛɥɚɫɬɢ ɨɩɪɟɞɟɥɟɧɢɹ ɮɭɧɤɰɢɣ

, ɪɹɞ ɦɨɠɟɬ ɫɯɨɞɢɬɶɫɹ, ɚ ɞɥɹ ɞɪɭɝɢɯ – ɪɚɫɯɨɞɢɬɶɫɹ.

()

xu

n

∞

=++++

......

¦

=

n

ɢɡ ɨɛɥɚɫɬɢ ɨɩɪɟɞɟɥɟɧɢɹ ɮɭɧɤɰɢɣ

x

0

++++ xuxuxu

00201

n

, (4.1)

xuxuxuxu

nn

1

, ɨɩɪɟɞɟɥɟɧɧɵɟ ɜ ɧɟ-

()

...,,21xuxu

xu

n

, (4.2)

......

x xx =

0

68

ɋɨɜɨɤɭɩɧɨɫɬɶ ɜɫɟɯ ɬɨɱɟɤ ɫɯɨɞɢɦɨɫɬɢ ɮɭɧɤɰɢɨɧɚɥɶɧɨɝɨ ɪɹɞɚ ɧɚɡɵɜɚɟɬɫɹ ɟɝɨ

x

ɨɛɥɚɫɬɶɸ ɫɯɨɞɢɦɨɫɬɢ.

ɑɚɫɬɢɱɧɨɣ ɫɭɦɦɨɣ ɮɭɧɤɰɢɨɧɚɥɶɧɨɝɨ ɪɹɞɚ ɧɚɡɵɜɚɟɬɫɹ ɫɭɦɦɚ ɟɝɨ

ɱɥɟɧɨɜ

() () () ()

21

+++= ...

xuxuxuxS

nn

,

ɹɜɥɹɸɳɚɹɫɹ ɮɭɧɤɰɢɟɣ ɩɟɪɟɦɟɧɧɨɣ

.

ɂɡ ɨɩɪɟɞɟɥɟɧɢɹ ɨɛɥɚɫɬɢ ɫɯɨɞɢɦɨɫɬɢ ɮɭɧɤɰɢɨɧɚɥɶɧɨɝɨ ɪɹɞɚ ɫɥɟɞɭɟɬ, ɱɬɨ ɞɥɹ

ɥɸɛɨɣ ɬɨɱɤɢ

ɝɞɟ

()

xS

ɷɬɨɣ ɨɛɥɚɫɬɢ ɫɭɳɟɫɬɜɭɟɬ ɩɪɟɞɟɥ ɱɚɫɬɢɱɧɨɣ ɫɭɦɦɵ

lim

n

() ()

n

∞→

,

xSxS

=

– ɫɭɦɦɚ ɮɭɧɤɰɢɨɧɚɥɶɧɨɝɨ ɪɹɞɚ.

n

ȼ ɬɨɱɤɚɯ, ɧɟ ɩɪɢɧɚɞɥɟɠɚɳɢɯ ɨɛɥɚɫɬɢ ɫɯɨɞɢɦɨɫɬɢ, ɱɚɫɬɢɱɧɚɹ ɫɭɦɦɚ ɧɟ ɢɦɟɟɬ ɩɪɟɞɟɥɚ. Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɫɭɦɦɚ ɟɬɫɹ ɧɟɤɨɬɨɪɨɣ ɮɭɧɤɰɢɟɣ ɩɟɪɟɦɟɧɧɨɣ

ɮɭɧɤɰɢɨɧɚɥɶɧɨɝɨ ɪɹɞɚ ɬɚɤɠɟ ɹɜɥɹ-

()

xS

, ɨɩɪɟɞɟɥɟɧɧɨɣ ɜ ɨɛɥɚɫɬɢ ɫɯɨɞɢɦɨɫɬɢ

ɪɹɞɚ.

ȿɫɥɢ ɮɭɧɤɰɢɨɧɚɥɶɧɵɣ ɪɹɞ (4.1) ɫɯɨɞɢɬɫɹ, ɬɨ ɜɵɩɨɥɧɹɟɬɫɹ ɪɚɜɟɧɫɬɜɨ

() () () ()

21

n

.

......

++++= xuxuxuxS

ȿɫɥɢ ɮɭɧɤɰɢɨɧɚɥɶɧɵɣ ɪɹɞ ɫɯɨɞɢɬɫɹ ɢ ɢɦɟɟɬ ɫɭɦɦɭ

() ()

ɧɚɡɵɜɚɟɬɫɹ ɟɝɨ ɨɫɬɚɬɤɨɦ ɩɨɫɥɟ

xSxSn−

ɨɛɨɡɧɚɱɚɬɶ ɱɟɪɟɡ

, ɨɧ ɩɨɥɭɱɟɧ ɢɡ ɪɹɞɚ (4.1) ɨɬɛɪɚɫɵɜɚɧɢɟɦ ɟɝɨ ɩɟɪɜɵɯ n

()

xr

n

-ɝɨ ɱɥɟɧɚ. Ɉɫɬɚɬɨɤ ɪɹɞɚ ɛɭɞɟɦ

, ɬɨ ɪɚɡɧɨɫɬɶ

()

xS

ɱɥɟɧɨɜ

() () () () () ()

21

......

+++

−=++++=

.

xSxSxuxuxuxr

nknnnn

Ɉɱɟɜɢɞɧɨ, ɱɬɨ

()

.

0lim =∞→xr

n

ɉɪɢɦɟɪ 4.1.

Ɉɩɪɟɞɟɥɢɬɶ ɨɛɥɚɫɬɶ ɫɯɨɞɢɦɨɫɬɢ ɮɭɧɤɰɢɨɧɚɥɶɧɨɝɨ ɪɹɞɚ

11

1

...

∞

1

=++++

...

¦

n

.

2242

nn

xxxx

=

1

Ɋɟɲɟɧɢɟ.

ɑɥɟɧɵ ɪɹɞɚ ɨɛɪɚɡɭɸɬ ɝɟɨɦɟɬɪɢɱɟɫɤɭɸ ɩɪɨɝɪɟɫɫɢɸ ɫɨ ɡɧɚɦɟɧɚɬɟɥɟɦ

Ɍɚɤ ɤɚɤ ɝɟɨɦɟɬɪɢɱɟɫɤɚɹ ɩɪɨɝɪɟɫɫɢɹ ɫɯɨɞɢɬɫɹ, ɟɫɥɢ

1<q

, ɢ ɪɚɫɯɨɞɢɬɫɹ, ɟɫɥɢ

n

-ɩɟɪɜɵɯ

()

xS

q =

()

n

1

xS

.

2

69

1≥q

x

, ɬɨ ɞɚɧɧɵɣ ɪɹɞ ɫɯɨɞɢɬɫɹ ɞɥɹ ɜɫɟɯ ɡɧɚɱɟɧɢɣ x, ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɯ ɧɟɪɚɜɟɧ-

1

ɫɬɜɭ

ɞɥɹ ɤɨɬɨɪɵɯ

ɥɨɜ

<

, ɢɥɢ

2

1−<<∞−

. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɚɧɧɵɣ ɪɹɞ ɫɯɨɞɢɬɫɹ ɞɥɹ ɜɫɟɯ ɬɨɱɟɤ x,

12>

1>x

. Ɉɛɥɚɫɬɶ ɫɯɨɞɢɦɨɫɬɢ ɞɚɧɧɨɝɨ ɪɹɞɚ ɫɨɫɬɨɢɬ ɢɡ ɞɜɭɯ ɢɧɬɟɪɜɚ-

ɢ +∞<<

1 . X

ɂɡɜɟɫɬɧɨ, ɱɬɨ: ɫɭɦɦɚ ɤɨɧɟɱɧɨɝɨ ɱɢɫɥɚ ɧɟɩɪɟɪɵɜɧɵɯ ɮɭɧɤɰɢɣ ɟɫɬɶ ɮɭɧɤɰɢɹ ɧɟɩɪɟɪɵɜɧɚɹ; ɩɪɨɢɡɜɨɞɧɚɹ ɢ ɢɧɬɟɝɪɚɥ ɨɬ ɫɭɦɦɵ ɤɨɧɟɱɧɨɝɨ ɱɢɫɥɚ ɮɭɧɤɰɢɣ ɪɚɜɧɵ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɫɭɦɦɟ ɩɪɨɢɡɜɨɞɧɵɯ ɢ ɢɧɬɟɝɪɚɥɨɜ ɨɬ ɷɬɢɯ ɮɭɧɤɰɢɣ. Ɉɞɧɚɤɨ, ɞɥɹ ɩɪɨɢɡɜɨɥɶɧɵɯ ɮɭɧɤɰɢɨɧɚɥɶɧɵɯ ɪɹɞɨɜ ɷɬɢ ɫɜɨɣɫɬɜɚ ɦɨɝɭɬ ɨɤɚɡɚɬɶɫɹ ɧɟɫɩɪɚɜɟɞɥɢɜɵɦɢ.

ɉɪɢɦɟɪ 4.2.

Ɉɩɪɟɞɟɥɢɬɶ ɫɭɦɦɭ ɪɹɞɚ

2

x

2

+

1

x

2

x

+

() () ()

+

1

...

2

x

2

x

++

2

+

1

x

∞

2

x

=+

...

¦

=

1

n

1

.

nn

2

+

x

Ɋɟɲɟɧɢɟ.

ɉɪɢ ɧɭɥɸ. ɉɪɢ

ɟɝɨ ɡɧɚɦɟɧɚɬɟɥɶ

0=

ɜɫɟ ɱɥɟɧɵ ɞɚɧɧɨɝɨ ɪɹɞɚ ɪɚɜɧɵ ɧɭɥɸ ɢ ɟɝɨ ɫɭɦɦɚ ɬɚɤɠɟ ɪɚɜɧɚ

0≠

ɪɹɞ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɫɯɨɞɹɳɢɣɫɹ ɝɟɨɦɟɬɪɢɱɟɫɤɢɣ, ɬɚɤ ɤɚɤ

1

=xq

<

, ɫɭɦɦɚ ɪɹɞɚ ɪɚɜɧɚ

2

1

+

2

x

a

1

S

=

q

1

−

§ ¨

©

2

x

+

1

−

1

+

1

=

1

x

.

· ¸

2

¹

()

xS

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɫɭɦɦɚ ɞɚɧɧɨɝɨ ɪɹɞɚ (ɨɧɚ ɢɦɟɟɬ ɪɚɡɪɵɜ ɜ ɬɨɱɤɟ

0=

):

ɹɜɥɹɟɬɫɹ ɪɚɡɪɵɜɧɨɣ ɮɭɧɤɰɢɟɣ

≠

;0ɟɫɥɢ,1

()

=

xS

 ® ¯

x

X

=

.0ɟɫɥɢ,0

x

4.2. ɉɪɚɜɢɥɶɧɨ ɫɯɨɞɹɳɢɟɫɹ

ɮɭɧɤɰɢɨɧɚɥɶɧɵɟ ɪɹɞɵ ɢ ɢɯ ɫɜɨɣɫɬɜɚ

Ɏɭɧɤɰɢɨɧɚɥɶɧɵɣ ɪɹɞ (4.1) ɧɚɡɵɜɚɟɬɫɹ

, ɟɫɥɢ ɫɭɳɟɫɬɜɭɟɬ ɬɚɤɨɣ ɡɧɚɤɨɩɨɥɨɠɢɬɟɥɶɧɵɣ ɫɯɨɞɹɳɢɣɫɹ ɱɢɫɥɨɜɨɣ ɪɹɞ

[]

ba,

ɩɪɚɜɢɥɶɧɨ ɫɯɨɞɹɳɢɦɫɹ ɧɚ ɨɬɪɟɡɤɟ

70