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Файл:Числовые и функциональные ряды. Учебник
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ȿ. Ƚ. ɉɥɨɬɧɢɤɨɜɚ, ȼ. ȼ. Ʌɨɝɢɧɨɜɚ
ɑɂɋɅɈȼɕȿ ɂ ɎɍɇɄɐɂɈɇȺɅɖɇɕȿ ɊəȾɕ
ɍɱɟɛɧɢɤ
Ɇɨɫɤɜɚ ȼɨɥɨɝɞɚ
«ɂɧɮɪɚ-ɂɧɠɟɧɟɪɢɹ»
2024
1
ɍȾɄ 517.11
ȻȻɄ 22.161
ɉ39
Ɋɟɰɟɧɡɟɧɬɵ:
ɞɨɤɬɨɪ ɩɟɞɚɝɨɝɢɱɟɫɤɢɯ ɧɚɭɤ, ɩɪɨɮɟɫɫɨɪ, ɡɚɜɟɞɭɸɳɢɣ ɤɚɮɟɞɪɨɣ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɚɧɚɥɢɡɚ,
ɬɟɨɪɢɢ ɢ ɦɟɬɨɞɢɤɢ ɨɛɭɱɟɧɢɹ ɦɚɬɟɦɚɬɢɤɟ ɎȽȻɈɍ ȼɈ «əɪɨɫɥɚɜɫɤɢɣ ɝɨɫɭɞɚɪɫɬɜɟɧɧɵɣ
ɩɟɞɚɝɨɝɢɱɟɫɤɢɣ ɭɧɢɜɟɪɫɢɬɟɬ ɢɦ. Ʉ. Ⱦ. ɍɲɢɧɫɤɨɝɨ» ȿ. ɂ. ɋɦɢɪɧɨɜ;
ɞɨɤɬɨɪ ɮɢɡɢɤɨ-ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɧɚɭɤ, ɩɪɨɮɟɫɫɨɪ, ɡɚɜɟɞɭɸɳɢɣ ɤɚɮɟɞɪɨɣ ɜɵɫɲɟɣ ɦɚɬɟɦɚɬɢɤɢ
ɎȽȺɈɍ ȼɈ «ɉɟɪɦɫɤɢɣ ɧɚɰɢɨɧɚɥɶɧɵɣ ɢɫɫɥɟɞɨɜɚɬɟɥɶɫɤɢɣ ɩɨɥɢɬɟɯɧɢɱɟɫɤɢɣ ɭɧɢɜɟɪɫɢɬɟɬ»
Ⱥ. Ɋ. Ⱥɛɞɭɥɥɚɟɜ
ɉɥɨɬɧɢɤɨɜɚ, ȿ. Ƚ.
ɉ39 ɑɢɫɥɨɜɵɟ ɢ ɮɭɧɤɰɢɨɧɚɥɶɧɵɟ ɪɹɞɵ : ɭɱɟɛɧɢɤ / ȿ. Ƚ. ɉɥɨɬɧɢɤɨɜɚ,
ȼ. ȼ. Ʌɨɝɢɧɨɜɚ. – Ɇɨɫɤɜɚ ; ȼɨɥɨɝɞɚ : ɂɧɮɪɚ-ɂɧɠɟɧɟɪɢɹ, 2024. – 172 ɫ. :
ɢɥ., ɬɚɛɥ.
ISBN 978-5-9729-1983-3
Ⱦɚɧɵ ɨɩɪɟɞɟɥɟɧɢɹ ɢ ɫɜɨɣɫɬɜɚ ɱɢɫɥɨɜɵɯ ɪɹɞɨɜ, ɩɪɢɡɧɚɤɢ ɫɯɨɞɢɦɨɫɬɢ ɱɢɫɥɨɜɵɯ
ɪɹɞɨɜ, ɨɩɪɟɞɟɥɟɧɢɹ ɢ ɫɜɨɣɫɬɜɚ ɮɭɧɤɰɢɨɧɚɥɶɧɵɯ ɪɹɞɨɜ, ɩɨɧɹɬɢɟ ɩɪɚɜɢɥɶɧɨ ɫɯɨɞɹɳɟɝɨɫɹ ɮɭɧɤɰɢɨɧɚɥɶɧɨɝɨ ɪɹɞɚ. ɉɪɟɞɫɬɚɜɥɟɧ ɬɟɨɪɟɬɢɱɟɫɤɢɣ ɦɚɬɟɪɢɚɥ ɫɨ ɫɬɪɨɝɢɦɢ ɦɚɬɟɦɚɬɢɱɟɫɤɢɦɢ ɮɨɪɦɭɥɢɪɨɜɤɚɦɢ ɢ ɞɨɤɚɡɚɬɟɥɶɫɬɜɚɦɢ, ɚ ɬɚɤɠɟ ɩɨɞɪɨɛɧɨ ɪɚɡɨɛɪɚɧɧɵɟ ɩɪɢɦɟɪɵ. Ʉɚɠɞɚɹ ɝɥɚɜɚ ɡɚɜɟɪɲɚɟɬɫɹ ɡɚɞɚɧɢɹɦɢ ɞɥɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɝɨ ɪɟɲɟɧɢɹ, ɜ ɤɨɧɰɟ
ɪɚɡɞɟɥɨɜ ɩɪɢɜɨɞɢɬɫɹ ɩɨ ɞɜɚ ɜɚɪɢɚɧɬɚ ɬɟɫɬɨɜ ɞɥɹ ɫɚɦɨɤɨɧɬɪɨɥɹ ɢ ɫɢɫɬɟɦɚɬɢɡɚɰɢɢ ɡɧɚɧɢɣ. ɂɦɟɸɬɫɹ ɨɬɜɟɬɵ ɤɨ ɜɫɟɦ ɡɚɞɚɧɢɹɦ, ɱɬɨ ɩɨɡɜɨɥɢɬ ɨɪɝɚɧɢɡɨɜɚɬɶ ɫɚɦɨɫɬɨɹɬɟɥɶɧɭɸ
ɪɚɛɨɬɭ ɫɬɭɞɟɧɬɨɜ ɩɪɢ ɢɡɭɱɟɧɢɢ ɭɱɟɛɧɨɝɨ ɦɚɬɟɪɢɚɥɚ ɪɚɡɞɟɥɨɜ.
Ⱦɥɹ ɛɚɤɚɥɚɜɪɨɜ, ɫɩɟɰɢɚɥɢɫɬɨɜ ɢ ɚɫɩɢɪɚɧɬɨɜ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ, ɢɧɠɟɧɟɪɧɨ-ɬɟɯɧɢɱɟɫɤɢɯ, ɷɤɨɧɨɦɢɱɟɫɤɢɯ ɧɚɩɪɚɜɥɟɧɢɣ ɜɭɡɨɜ, ɩɨɞɪɨɛɧɨ ɢɡɭɱɚɸɳɢɯ ɦɚɬɟɦɚɬɢɱɟɫɤɢɣ ɚɧɚɥɢɡ, ɚ ɬɚɤɠɟ ɞɥɹ ɩɪɟɩɨɞɚɜɚɬɟɥɟɣ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɞɢɫɰɢɩɥɢɧ.
ɍȾɄ 517.11
ȻȻɄ 22.161
ISBN 978-5-9729-1983-3 © ɉɥɨɬɧɢɤɨɜɚ ȿ. Ƚ., Ʌɨɝɢɧɨɜɚ ȼ. ȼ., 2024
© ɂɡɞɚɬɟɥɶɫɬɜɨ «ɂɧɮɪɚ-ɂɧɠɟɧɟɪɢɹ», 2024
© Ɉɮɨɪɦɥɟɧɢɟ. ɂɡɞɚɬɟɥɶɫɬɜɨ «ɂɧɮɪɚ-ɂɧɠɟɧɟɪɢɹ», 2024
2
ɋɈȾȿɊɀȺɇɂȿ
ɉɪɟɞɢɫɥɨɜɢɟ ............................................................................................................. 5
ȼɜɟɞɟɧɢɟ .................................................................................................................... 7
ɊȺɁȾȿɅ I. ɑɂɋɅɈȼɕȿ ɊəȾɕ ........................................................................... 8
Ƚɥɚɜɚ 1. Ɉɛɳɢɟ ɨɩɪɟɞɟɥɟɧɢɹ ɢ ɫɜɨɣɫɬɜɚ ɱɢɫɥɨɜɵɯ ɪɹɞɨɜ ............................. 8
1.1. Ɉɩɪɟɞɟɥɟɧɢɟ ɢ ɩɪɢɦɟɪɵ ɱɢɫɥɨɜɨɝɨ ɪɹɞɚ ......................................................... 8
1.2. ɋɜɨɣɫɬɜɚ ɱɢɫɥɨɜɵɯ ɪɹɞɨɜ ................................................................................ 13
1.3. ɇɟɨɛɯɨɞɢɦɵɣ ɩɪɢɡɧɚɤ ɫɯɨɞɢɦɨɫɬɢ ................................................................ 17
Ɂɚɞɚɧɢɹ ɞɥɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɝɨ ɪɟɲɟɧɢɹ ............................................................. 21
Ƚɥɚɜɚ 2. ɂɫɫɥɟɞɨɜɚɧɢɟ ɫɯɨɞɢɦɨɫɬɢ ɱɢɫɥɨɜɵɯ ɪɹɞɨɜ
ɫ ɩɨɥɨɠɢɬɟɥɶɧɵɦɢ ɱɥɟɧɚɦɢ ............................................................................... 23
2.1. ɍɫɥɨɜɢɟ ɫɯɨɞɢɦɨɫɬɢ ɱɢɫɥɨɜɨɝɨ ɪɹɞɚ ............................................................. 23
2.2. ɉɪɢɡɧɚɤɢ ɫɪɚɜɧɟɧɢɹ ......................................................................................... 25
2.3. Ⱦɨɫɬɚɬɨɱɧɵɟ ɩɪɢɡɧɚɤɢ Ⱦɚɥɚɦɛɟɪɚ,
Ɂɚɞɚɧɢɹ ɞɥɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɝɨ ɪɟɲɟɧɢɹ ............................................................. 36
Ƚɥɚɜɚ 3. ɂɫɫɥɟɞɨɜɚɧɢɟ ɫɯɨɞɢɦɨɫɬɢ ɡɧɚɤɨɩɟɪɟɦɟɧɧɵɯ ɱɢɫɥɨɜɵɯ ɪɹɞɨɜ ... 39
3.1. Ɂɧɚɤɨɱɟɪɟɞɭɸɳɢɟɫɹ ɱɢɫɥɨɜɵɟ ɪɹɞɵ .............................................................. 39
3.2. Ⱥɛɫɨɥɸɬɧɚɹ ɢ ɭɫɥɨɜɧɚɹ ɫɯɨɞɢɦɨɫɬɶ ɡɧɚɤɨɩɟɪɟɦɟɧɧɵɯ
ɱɢɫɥɨɜɵɯ ɪɹɞɨɜ ........................................................................................................ 43
Ɂɚɞɚɧɢɹ ɞɥɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɝɨ ɪɟɲɟɧɢɹ ............................................................. 47
Ɍɟɫɬɵ ɞɥɹ ɫɚɦɨɤɨɧɬɪɨɥɹ ɢ ɫɢɫɬɟɦɚɬɢɡɚɰɢɢ ɡɧɚɧɢɣ ...................................... 48
ɊȺɁȾȿɅ II. ɎɍɇɄɐɂɈɇȺɅɖɇɕȿ ɊəȾɕ ..................................................... 68
Ƚɥɚɜɚ 4. Ɏɭɧɤɰɢɨɧɚɥɶɧɵɟ ɪɹɞɵ: ɨɛɳɢɟ ɨɩɪɟɞɟɥɟɧɢɹ ɢ ɫɜɨɣɫɬɜɚ ............. 68
4.1. Ɉɛɥɚɫɬɶ ɫɯɨɞɢɦɨɫɬɢ ɮɭɧɤɰɢɨɧɚɥɶɧɨɝɨ ɪɹɞɚ
4.2. ɉɪɚɜɢɥɶɧɨ ɫɯɨɞɹɳɢɟɫɹ ɮɭɧɤɰɢɨɧɚɥɶɧɵɟ ɪɹɞɵ ɢ ɢɯ ɫɜɨɣɫɬɜɚ .................... 70
Ɂɚɞɚɧɢɹ ɞɥɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɝɨ ɪɟɲɟɧɢɹ ............................................................. 75
Ƚɥɚɜɚ 5. ɋɬɟɩɟɧɧɵɟ ɪɹɞɵ .................................................................................... 78
5.1. Ɉɩɪɟɞɟɥɟɧɢɟ, ɨɛɥɚɫɬɶ ɢ ɪɚɞɢɭɫ ɫɯɨɞɢɦɨɫɬɢ ɫɬɟɩɟɧɧɨɝɨ ɪɹɞɚ .................... 78
5.2. ȼɵɱɢɫɥɟɧɢɟ ɪɚɞɢɭɫɚ ɢ ɢɫɫɥɟɞɨɜɚɧɢɟ ɨɛɥɚɫɬɢ ɫɯɨɞɢɦɨɫɬɢ
ɫɬɟɩɟɧɧɨɝɨ ɪɹɞɚ ....................................................................................................... 81
5.3. ɋɜɨɣɫɬɜɚ ɫɬɟɩɟɧɧɵɯ ɪɹɞɨɜ .............................................................................. 85
Ɂɚɞɚɧɢɹ ɞɥɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɝɨ ɪɟɲɟɧɢɹ ............................................................. 86
Ƚɥɚɜɚ 6. Ɋɚɡɥɨɠɟɧɢɟ ɮɭɧɤɰɢɣ ɜ ɫɬɟɩɟɧɧɵɟ ɪɹɞɵ .......................................... 87
6.1. Ɋɹɞɵ Ɍɟɣɥɨɪɚ ɢ Ɇɚɤɥɨɪɟɧɚ ............................................................................. 87
6.2.
Ɋɚɡɥɨɠɟɧɢɟ ɧɟɤɨɬɨɪɵɯ ɷɥɟɦɟɧɬɚɪɧɵɯ ɮɭɧɤɰɢɣ ɜ ɪɹɞ Ɇɚɤɥɨɪɟɧɚ ............. 93
6.3. Ɋɚɡɥɨɠɟɧɢɟ ɮɭɧɤɰɢɢ ɜ ɫɬɟɩɟɧɧɨɣ ɪɹɞ ɫ ɩɨɦɨɳɶɸ ɢɡɜɟɫɬɧɵɯ
ɪɚɡɥɨɠɟɧɢɣ ɢ ɫɜɨɣɫɬɜ ............................................................................................. 98
Ɂɚɞɚɧɢɹ ɞɥɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɝɨ ɪɟɲɟɧɢɹ ........................................................... 103
Ƚɥɚɜɚ 7. ɉɪɢɥɨɠɟɧɢɟ ɱɢɫɥɨɜɵɯ ɪɹɞɨɜ ɤ ɩɪɢɛɥɢɠɟɧɧɵɦ
ɜɵɱɢɫɥɟɧɢɹɦ ........................................................................................................ 105
7.1. ȼɵɱɢɫɥɟɧɢɟ ɡɧɚɱɟɧɢɣ ɮɭɧɤɰɢɣ .................................................................... 105
7.2. ɉɪɢɛɥɢɠɟɧɧɨɟ ɜɵɱɢɫɥɟɧɢɟ ɢɧɬɟɝɪɚɥɨɜ ....................................................... 108
Ʉɨɲɢ, Ʉɨɲɢ – Ɇɚɤɥɨɪɟɧɚ .................. 29
................................................ 68
3
7.3. ɉɪɢɛɥɢɠɟɧɧɨɟ ɪɟɲɟɧɢɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ .......................... 111
Ɂɚɞɚɧɢɹ ɞɥɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɝɨ ɪɟɲɟɧɢɹ ........................................................... 115
Ƚɥɚɜɚ 8. Ɋɹɞɵ Ɏɭɪɶɟ ........................................................................................... 117
8.1. Ɍɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɢɟ ɪɹɞɵ. Ɋɹɞɵ Ɏɭɪɶɟ ...................................................... 117
8.2. Ɋɚɡɥɨɠɟɧɢɟ ɜ ɪɹɞ Ɏɭɪɶɟ ɱɟɬɧɵɯ ɢ ɧɟɱɟɬɧɵɯ ɮɭɧɤɰɢɣ .............................. 128
8.3. Ɋɹɞ Ɏɭɪɶɟ ɜ ɩɪɨɢɡɜɨɥɶɧɨɦ ɢɧɬɟɪɜɚɥɟ .......................................................... 132
Ɂɚɞɚɧɢɹ ɞɥɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɝɨ ɪɟɲɟɧɢɹ ........................................................... 136
Ɍɟɫɬɵ ɞɥɹ ɫɚɦɨɤɨɧɬɪɨɥɹ ɢ ɫɢɫɬɟɦɚɬɢɡɚɰɢɢ ɡɧɚɧɢɣ .................................... 137
Ɉɬɜɟɬɵ ɤ ɡɚɞɚɧɢɹɦ ɞɥɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɝɨ ɪɟɲɟɧɢɹ ................................... 155
Ɉɬɜɟɬɵ ɤ
Ʌɢɬɟɪɚɬɭɪɚ ............................................................................................................ 167
ɬɟɫɬɚɦ .................................................................................................. 166
4
ɉɊȿȾɂɋɅɈȼɂȿ
Ⱦɚɧɧɵɣ ɭɱɟɛɧɢɤ ɫɨɫɬɨɢɬ ɢɡ ɞɜɭɯ ɪɚɡɞɟɥɨɜ: «ɑɢɫɥɨɜɵɟ ɪɹɞɵ» ɢ «Ɏɭɧɤɰɢɨɧɚɥɶɧɵɟ ɪɹɞɵ». ȿɝɨ ɫɨɞɟɪɠɚɧɢɟ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɩɪɨɝɪɚɦɦɟ ɩɨɞɝɨɬɨɜɤɢ ɛɚɤɚɥɚɜɪɨɜ ɢ ɫɩɟɰɢɚɥɢɫɬɨɜ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ, ɢɧɠɟɧɟɪɧɨ-ɬɟɯɧɢɱɟɫɤɢɯ, ɷɤɨɧɨɦɢɱɟɫɤɢɯ
ɧɚɩɪɚɜɥɟɧɢɣ ɜɭɡɨɜ, ɩɨɞɪɨɛɧɨ ɢɡɭɱɚɸɳɢɯ ɦɚɬɟɦɚɬɢɱɟɫɤɢɣ ɚɧɚɥɢɡ.
Ɋɚɡɞɟɥ «ɑɢɫɥɨɜɵɟ ɪɹɞɵ» ɫɨɞɟɪɠɢɬ ɫɥɟɞɭɸɳɢɟ ɝɥɚɜɵ: Ɉɛɳɢɟ ɨɩɪɟɞɟɥɟɧɢɹ
ɢ ɫɜɨɣɫɬɜɚ ɱɢɫɥɨɜɵɯ ɪɹɞɨɜ; ɂɫɫɥɟɞɨɜɚɧɢɟ ɫɯɨɞɢɦɨɫɬɢ ɱɢɫɥɨɜɵɯ ɪɹɞɨɜ ɫ ɩɨɥɨɠɢɬɟɥɶɧɵɦɢ
ɪɹɞɨɜ. ȼ ɪɚɡɞɟɥ «Ɏɭɧɤɰɢɨɧɚɥɶɧɵɟ ɪɹɞɵ» ɜɨɲɥɢ ɝɥɚɜɵ: Ɏɭɧɤɰɢɨɧɚɥɶɧɵɟ ɪɹɞɵ:
ɨɛɳɢɟ ɨɩɪɟɞɟɥɟɧɢɹ ɢ ɫɜɨɣɫɬɜɚ; ɋɬɟɩɟɧɧɵɟ ɪɹɞɵ; Ɋɚɡɥɨɠɟɧɢɟ ɮɭɧɤɰɢɣ ɜ ɫɬɟɩɟɧɧɵɟ ɪɹɞɵ; ɉɪɢɥɨɠɟɧɢɟ ɱɢɫɥɨɜɵɯ ɪɹɞɨɜ ɤ ɩɪɢɛɥɢɠɟɧɧɵɦ ɜɵɱɢɫɥɟɧɢɹɦ; Ɋɹɞɵ
Ɏɭɪɶɟ.
ȼ ɪɟɡɭɥɶɬɚɬɟ ɢɡɭɱɟɧɢɹ ɦɚɬɟɪɢɚɥɚ ɭɱɟɛɧɢɤɚ ɨɛɭɱɚɸɳɢɟɫɹ ɞɨɥɠɧɵ:
ɡɧɚɬɶ
ɨɩɪɟɞɟɥɟɧɢɹ ɢ ɫɜɨɣɫɬɜɚ ɱɢɫɥɨɜɵɯ ɪɹɞɨɜ; ɩɪɢɡɧɚɤɢ ɫɯɨɞɢɦɨɫɬɢ ɱɢɫɥɨɜɵɯ
ɪɹɞɨɜ; ɨɩɪɟɞɟɥɟɧɢɹ ɢ ɫɜɨɣɫɬɜɚ ɮɭɧɤɰɢɨɧɚɥɶɧɵɯ ɪɹɞɨɜ; ɩɨɧɹɬɢɟ ɩɪɚɜɢɥɶɧɨ ɫɯɨɞɹɳɟɝɨɫɹ ɮɭɧɤɰɢɨɧɚɥɶɧɨɝɨ ɪɹɞɚ;
ɭɦɟɬɶ
ɢɫɫɥɟɞɨɜɚɬɶ ɫɯɨɞɢɦɨɫɬɶ ɱɢɫɥɨɜɵɯ ɪɚɞɨɜ; ɨɩɪɟɞɟɥɹɬɶ ɪɚɞɢɭɫ, ɢɧɬɟɪɜɚɥ ɢ ɨɛɥɚɫɬɶ ɫɯɨɞɢɦɨɫɬɢ ɫɬɟɩɟɧɧɨɝɨ ɪɹɞɚ; ɨɫɭɳɟɫɬɜɥɹɬɶ ɪɚɡɥɨɠɟɧɢɟ ɮɭɧɤɰɢɣ ɜ ɪɹɞɵ
Ɍɟɣɥɨɪɚ ɢ Ɇɚɤɥɨɪɟɧɚ; ɨɫɭɳɟɫɬɜɥɹɬɶ ɪɚɡɥɨɠɟɧɢɟ ɮɭɧɤɰɢɣ ɜ ɪɹɞ Ɏɭɪɶɟ;
ɜɥɚɞɟɬɶ
ɧɚɜɵɤɚɦɢ ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɪɹɞɨɜ ɤ ɩɪɢɛɥɢɠɟɧɧɵɦ ɜɵɱɢɫɥɟɧɢɹɦ.
Ⱦɥɹ ɭɫɩɟɲɧɨɝɨ ɨɜɥɚɞɟɧɢɹ ɭɱɟɛɧɵɦ
ɡɧɚɬɶ ɬɟɨɪɢɸ ɩɪɟɞɟɥɨɜ ɢ ɦɟɬɨɞɢɤɭ ɢɯ ɜɵɱɢɫɥɟɧɢɹ; ɜɥɚɞɟɬɶ ɭɦɟɧɢɹɦɢ ɢ ɧɚɜɵɤɚɦɢ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɢ ɢɧɬɟɝɪɚɥɶɧɨɝɨ ɢɫɱɢɫɥɟɧɢɹ.
ȼ ɬɟɨɪɟɬɢɱɟɫɤɨɣ ɱɚɫɬɢ ɝɥɚɜ ɭɱɟɛɧɢɤɚ ɢɡɥɨɠɟɧɢɟ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɫɬɪɨɝɨ ɦɚɬɟɦɚɬɢɱɟɫɤɢ ɫ ɜɵɜɨɞɨɦ ɜɫɟɯ ɩɨɥɨɠɟɧɢɣ ɢ ɞɨɤɚɡɚɬɟɥɶɫɬɜɨɦ ɬɟɨɪɟɦ. Ɍɟɨɪɢɹ ɫɨɩɪɨɜɨɠɞɚɟɬɫɹ ɦɧɨɝɨɱɢɫɥɟɧɧɵɦɢ ɩɪɢɦɟɪɚɦɢ, ɢɥɥɸɫɬɪɢɪɭɸɳɢɦɢ ɦɚɬɟɦɚɬɢɱɟɫɤɢɟ ɩɨɧɹɬɢɹ ɢ ɦɟɬɨɞɵ ɢɫɫɥɟɞɨɜɚɧɢɹ ɪɹɞɨɜ. Ɍɚɦ,
ɚɥɝɨɪɢɬɦɵ ɢɫɫɥɟɞɨɜɚɧɢɹ.
Ʉɚɠɞɚɹ ɝɥɚɜɚ ɡɚɜɟɪɲɚɟɬɫɹ ɡɚɞɚɧɢɹɦɢ ɞɥɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɝɨ ɪɟɲɟɧɢɟ, ɤɨɥɢɱɟɫɬɜɨ ɤɨɬɨɪɵɯ ɞɨɫɬɚɬɨɱɧɨ ɞɥɹ ɨɪɝɚɧɢɡɚɰɢɢ ɚɭɞɢɬɨɪɧɨɣ ɢ ɞɨɦɚɲɧɟɣ ɪɚɛɨɬɵ ɫɬɭɞɟɧɬɨɜ. ɇɚɩɪɢɦɟɪ, ɡɚɞɚɧɢɹ ɫ ɧɟɱɟɬɧɵɦɢ ɧɨɦɟɪɚɦɢ ɦɨɝɭɬ ɛɵɬɶ ɪɚɡɨɛɪɚɧɵ ɧɚ ɡɚɧɹɬɢɢ ɜ ɚɭɞɢɬɨɪɢɢ, ɚ ɡɚɞɚɧɢɹ ɫ ɱɟɬɧɵɦɢ ɧɨɦɟɪɚɦɢ ɩɪɟɞɥɨɠɟɧɵ ɫɬɭɞɟɧɬɚɦ ɞɥɹ
ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɣ ɞɨɦɚɲɧɟɣ ɪɚɛɨɬɵ. ȼ ɤɨɧɰɟ
ɡɚɞɚɧɢɹɦ, ɱɬɨ ɩɨɡɜɨɥɢɬ ɨɛɭɱɚɟɦɵɦ ɩɪɨɜɟɪɢɬɶ ɩɪɚɜɢɥɶɧɨɫɬɶ ɜɵɩɨɥɧɟɧɧɨɣ ɪɚɛɨɬɵ.
Ɋɚɡɞɟɥɵ ɭɱɟɛɧɢɤɚ ɡɚɜɟɪɲɚɸɬɫɹ ɞɜɭɦɹ ɜɚɪɢɚɧɬɚɦɢ ɬɟɫɬɨɜ ɞɥɹ ɤɨɧɬɪɨɥɹ ɢ ɫɢɫɬɟɦɚɬɢɡɚɰɢɢ ɩɨɥɭɱɟɧɧɵɯ ɡɧɚɧɢɣ, ɭɦɟɧɢɣ ɢ ɧɚɜɵɤɨɜ. ȼ ɤɚɠɞɨɦ ɬɟɫɬɟ 30 ɡɚɞɚɧɢɣ
ɡɚɤɪɵɬɨɝɨ ɬɢɩɚ ɪɚɡɧɨɝɨ ɭɪɨɜɧɹ ɫɥɨɠɧɨɫɬɢ. ȼɫɟ ɡɚɞɚɧɢɹ ɧɚ ɨɞɢɧɨɱɧɵɣ ɜɵɛɨɪ: ɢɡ
ɩɪɟɞɥɨɠɟɧɧɵɯ ɩɹɬɢ ɜɚɪɢɚɧɬɨɜ ɨɬɜɟɬɨɜ ɩɪɚɜɢɥɶɧɵɦ
ɜɟɬɵ ɧɚ ɡɚɞɚɧɢɹ ɬɟɫɬɨɜ ɬɚɤɠɟ ɩɪɢɜɟɞɟɧɵ ɜ ɤɨɧɰɟ ɭɱɟɛɧɢɤɚ, ɬɟɦ ɫɚɦɵɦ ɫɬɭɞɟɧɬɵ
ɱɥɟɧɚɦɢ; ɂɫɫɥɟɞɨɜɚɧɢɟ ɫɯɨɞɢɦɨɫɬɢ ɡɧɚɤɨɩɟɪɟɦɟɧɧɵɯ ɱɢɫɥɨɜɵɯ
ɦɚɬɟɪɢɚɥɨɦ ɨɛɭɱɚɸɳɢɟɫɹ ɞɨɥɠɧɵ:
ɝɞɟ ɷɬɨ ɜɨɡɦɨɠɧɨ, ɩɪɢɜɨɞɹɬɫɹ
ɭɱɟɛɧɢɤɚ ɩɪɢɜɟɞɟɧɵ ɨɬɜɟɬɵ ɤɨ ɜɫɟɦ
ɹɜɥɹɟɬɫɹ ɬɨɥɶɤɨ ɨɞɢɧ. Ɉɬ-
5
ɦɨɝɭɬ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨ ɨɰɟɧɢɬɶ ɪɟɡɭɥɶɬɚɬ ɫɜɨɟɝɨ ɪɟɲɟɧɢɹ. ɇɚ ɨɫɧɨɜɟ ɞɚɧɧɵɯ ɬɟɫɬɨɜ ɩɪɟɩɨɞɚɜɚɬɟɥɢ ɞɢɫɰɢɩɥɢɧɵ ɦɨɝɭɬ ɫɨɫɬɚɜɢɬɶ ɜɚɪɢɚɧɬɵ ɩɪɨɜɟɪɨɱɧɵɯ ɪɚɛɨɬ
ɪɚɡɧɨɝɨ ɭɪɨɜɧɹ ɫɥɨɠɧɨɫɬɢ ɢ ɨɛɴɟɦɚ.
ɍɤɚɡɚɧɧɵɟ ɨɫɨɛɟɧɧɨɫɬɢ ɭɱɟɛɧɢɤɚ ɩɨɡɜɨɥɹɸɬ ɢɫɩɨɥɶɡɨɜɚɬɶ ɟɝɨ ɜ ɨɪɝɚɧɢɡɚɰɢɢ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɣ ɪɚɛɨɬɵ ɫɬɭɞɟɧɬɨɜ, ɜ ɬɨɦ ɱɢɫɥɟ ɜ ɮɨɪɦɚɬɟ ɞɢɫɬɚɧɰɢɨɧɧɨɝɨ
ɨɛɭɱɟɧɢɹ.
6
ȼȼȿȾȿɇɂȿ
ɍɠɟ ɜ Ⱦɪɟɜɧɟɣ Ƚɪɟɰɢɢ ɭɱɟɧɵɟ ɢɫɩɨɥɶɡɨɜɚɥɢ ɛɟɫɤɨɧɟɱɧɵɟ ɫɭɦɦɵ ɞɥɹ
ɧɚɯɨɠɞɟɧɢɹ ɩɥɨɳɚɞɟɣ ɮɢɝɭɪ, ɨɛɴɟɦɨɜ ɬɟɥ, ɞɥɢɧ ɤɪɢɜɵɯ ɢ ɬ. ɞ. ɇɚɩɪɢɦɟɪ, Ⱥɪɯɢɦɟɞ ɞɥɹ ɜɵɱɢɫɥɟɧɢɹ ɩɥɨɳɚɞɢ ɩɚɪɚɛɨɥɢɱɟɫɤɨɝɨ ɫɟɝɦɟɧɬɚ (ɮɢɝɭɪɵ, ɨɝɪɚɧɢɱɟɧɧɨɣ
ɩɪɹɦɨɣ ɢ ɩɚɪɚɛɨɥɨɣ) ɧɚɲɟɥ ɫɭɦɦɭ ɛɟɫɤɨɧɟɱɧɨɣ ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɩɪɨɝɪɟɫɫɢɢ ɫɨ
ɡɧɚɦɟɧɚɬɟɥɟɦ 1/4.
Ʉɚɤ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɟ ɩɨɧɹɬɢɟ, ɪɹɞ ɦɚɬɟɦɚɬɢɤɢ ɫɬɚɥɢ ɢɫɩɨɥɶɡɨɜɚɬɶ ɜ
XVII ɜɟɤɟ. ɂɫɚɚɤ
ɝɟɛɪɚɢɱɟɫɤɢɯ ɢ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ. ȼ XVIII–XIX ɜɟɤɚɯ ɬɟɨɪɢɹ ɪɹɞɨɜ ɪɚɡɜɢɜɚɥɚɫɶ ɜ ɪɚɛɨɬɚɯ əɤɨɛɚ ɢ ɂɨɝɚɧɧɚ Ȼɟɪɧɭɥɥɢ, Ȼɪɭɤɚ Ɍɟɣɥɨɪɚ, Ʉɨɥɢɧɚ
Ɇɚɤɥɨɪɟɧɚ, Ʌɟɨɧɚɪɞɨ ɗɣɥɟɪɚ, ɀɚɧɚ Ⱦɚɥɚɦɛɟɪɚ, ɀɨɡɟɮɚ Ʌɚɝɪɚɧɠɚ ɢ ɞɪ. ɋɬɪɨɝɚɹ ɬɟɨɪɢɹ ɪɹɞɨɜ ɛɵɥɚ ɫɨɡɞɚɧɚ ɜ XIX ɜɟɤɟ ɧɚ ɨɫɧɨɜɟ ɩɨɧɹɬɢɹ ɩɪɟɞɟɥɚ ɜ ɬɪɭɞɚɯ
Ʉɚɪɥɚ Ƚɚɭɫɫɚ, Ȼɟɪɧɚɪɞɚ Ȼɨɥɶɰɚɧɨ, Ɉɝɸɫɬɟɧɚ Ʉɨɲɢ, ɉɟɬɟɪɚ Ⱦɢɪɢɯɥɟ, ɇɢɥɶɫɚ
Ⱥɛɟɥɹ, Ʉɚɪɥɚ ȼɟɣɟɪɲɬɪɚɫɫɚ, Ȼɟɪɧɯɚɪɞɚ Ɋɢɦɚɧɚ ɢ ɞɪ.
ȼ ɧɚɫɬɨɹɳɟɟ ɜɪɟɦɹ ɪɹɞɵ ɲɢɪɨɤɨ ɢɫɩɨɥɶɡɭɸɬɫɹ ɩɪɢ ɪɟɲɟɧɢɢ ɫɚɦɵɯ ɪɚɡɧɨɨɛɪɚɡɧɵɯ ɡɚɞɚɱ. ɉɪɟɠɞɟ ɜɫɟɝɨ, ɞɥɹ ɩɪɢɛɥɢɠɟɧɧɵɯ ɜɵɱɢɫɥɟɧɢɣ, ɞɥɹ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɮɭɧɤɰɢɣ, ɨɩɢɫɵɜɚɸɳɢɯ ɢɡɭɱɚɟɦɵɟ ɩɪɨɰɟɫɫɵ ɢɥɢ ɹɜɥɟɧɢɹ. ɗɬɢɦ ɢ ɨɛɴɹɫɧɹɟɬɫɹ ɨɛɹɡɚɬɟɥɶɧɨɟ ɢɡɭɱɟɧɢɟ ɬɟɨɪɢɢ ɱɢɫɥɨɜɵɯ
ɬɚɦɢ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ, ɢɧɠɟɧɟɪɧɨ-ɬɟɯɧɢɱɟɫɤɢɯ, ɷɤɨɧɨɦɢɱɟɫɤɢɯ ɧɚɩɪɚɜɥɟɧɢɣ
ɩɨɞɝɨɬɨɜɤɢ.
ɇɶɸɬɨɧ ɢ Ƚɨɬɮɪɢɞ Ʌɟɣɛɧɢɰ ɩɪɢɦɟɧɹɥɢ ɪɹɞɵ ɞɥɹ ɪɟɲɟɧɢɹ ɚɥ-
ɢ ɮɭɧɤɰɢɨɧɚɥɶɧɵɯ ɪɹɞɨɜ ɫɬɭɞɟɧ-
7
Ɋɚɡɞɟɥ I
N
ɑɂɋɅɈȼɕȿ ɊəȾɕ
Ɋɚɡɞɟɥ «ɑɢɫɥɨɜɵɟ ɪɹɞɵ» ɹɜɥɹɟɬɫɹ ɨɞɧɢɦ ɢɡ ɧɚɢɛɨɥɟɟ ɜɚɠɧɵɯ ɢ ɢɧɬɟɪɟɫɧɵɯ
ɪɚɡɞɟɥɨɜ ɜɭɡɨɜɫɤɨɝɨ ɤɭɪɫɚ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɚɧɚɥɢɡɚ, ɤɚɤ ɫ ɬɟɨɪɟɬɢɱɟɫɤɨɣ, ɬɚɤ ɢ
ɫ ɩɪɚɤɬɢɱɟɫɤɨɣ ɬɨɱɤɢ ɡɪɟɧɢɹ. ȼ ɷɬɨɦ ɪɚɡɞɟɥɟ ɨɛɨɛɳɚɸɬɫɹ ɪɚɧɟɟ ɪɚɫɫɦɨɬɪɟɧɧɵɟ
ɩɨɧɹɬɢɹ ɢ ɦɟɬɨɞɵ ɤɭɪɫɚ, ɚ ɬɚɤɠɟ ɩɪɟɞɥɚɝɚɟɬɫɹ ɧɨɜɵɣ ɢɧɫɬɪɭɦɟɧɬ ɪɟɲɟɧɢɹ ɫɚɦɵɯ
ɪɚɡɧɨɨɛɪɚɡɧɵɯ ɡɚɞɚɱ.
Ƚɥɚɜɚ 1
ɈȻɓɂȿ ɈɉɊȿȾȿɅȿɇɂə
ɂ ɋȼɈɃɋɌȼȺ ɑɂɋɅɈȼɕɏ ɊəȾɈȼ
ɂɡɭɱɟɧɢɟ ɬɟɨɪɢɢ ɪɹɞɨɜ ɦɵ ɧɚɱɧɟɦ ɫ ɜɜɟɞɟɧɢɹ ɨɛɳɢɯ ɩɨɧɹɬɢɣ ɞɥɹ ɱɢɫɥɨɜɵɯ ɪɹɞɨɜ, ɪɚɫɫɦɨɬɪɢɦ ɢɯ ɩɪɢɦɟɪɵ, ɫɜɨɣɫɬɜɚ ɢ ɧɟɨɛɯɨɞɢɦɵɣ ɩɪɢɡɧɚɤ ɫɯɨɞɢɦɨɫɬɢ.
1.1. Ɉɩɪɟɞɟɥɟɧɢɟ ɢ ɩɪɢɦɟɪɵ ɱɢɫɥɨɜɨɝɨ ɪɹɞɚ
ɉɭɫɬɶ ɞɚɧɚ ɱɢɫɥɨɜɚɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ
ɜɚɟɬɫɹ ɛɟɫɤɨɧɟɱɧɚɹ ɫɭɦɦɚ ɱɥɟɧɨɜ ɞɚɧɧɨɣ ɱɢɫɥɨɜɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɜɢɞɚ
321
=+++++
......
ɡɞɟɫɶ ɱɢɫɥɚ
321 n
ɧɚɡɵɜɚɸɬɫɹ ɱɥɟɧɚɦɢ ɪɹɞɚ, ɚ ɱɥɟɧ
...,...,,,
aaaa
ɧɵɦ ɧɨɦɟɪɨɦ ɧɚɡɵɜɚɟɬɫɹ ɨɛɳɢɦ ɱɥɟɧɨɦ ɪɹɞɚ.
ɉɪɚɜɢɥɨ, ɡɚɩɢɫɚɧɧɨɟ ɜ ɜɢɞɟ ɮɨɪɦɭɥɵ ɢ ɩɨɡɜɨɥɹɸɳɟɟ ɤɚɠɞɨɦɭ ɧɨɦɟɪɭ
n ∈ ɩɨɫɬɚɜɢɬɶ ɜ ɫɨɨɬɜɟɬɫɬɜɢɟ ɱɥɟɧ
ɪɹɞɚ (1.1), ɧɚɡɵɜɚɟɬɫɹ ɮɨɪɦɭɥɨɣ ɨɛ-
a
n
ɳɟɝɨ ɱɥɟɧɚ ɪɹɞɚ. Ɋɹɞ ɫɱɢɬɚɟɬɫɹ ɡɚɞɚɧɧɵɦ, ɟɫɥɢ ɡɚɞɚɧ ɟɝɨ ɨɛɳɢɣ ɱɥɟɧ.
ɉɪɢɦɟɪ 1.1. Ⱦɚɧɚ ɮɨɪɦɭɥɚ ɨɛɳɟɝɨ ɱɥɟɧɚ ɱɢɫɥɨɜɨɝɨ ɪɹɞɚ
a
n
n
()( )
= ,
121
n
−⋅−
!
n
ɡɚɩɢɫɚɬɶ ɟɝɨ ɩɟɪɜɵɟ ɩɹɬɶ ɱɥɟɧɨɜ.
. ɑɢɫɥɨɜɵɦ ɪɹɞɨɦ ɧɚɡɵ-
{}
aA =
n
∞
, (1.1)
aaaaa
nn
¦
1
n
=
ɫ ɩɪɨɢɡɜɨɥɶ-
a
n
8
Ɋɟɲɟɧɢɟ.
ɉɨɞɫɬɚɜɥɹɟɦ ɜ ɮɨɪɦɭɥɭ ɨɛɳɟɝɨ ɱɥɟɧɚ ɧɨɦɟɪ
n
ɢ ɧɚɯɨɞɢɦ ɫɨɨɬɜɟɬɫɬɜɭɸ-
ɳɢɣ ɱɥɟɧ ɪɹɞɚ:
1
()( )
=a ;
1
1121
−⋅⋅−
1
!1
−=
2
2
()( )
=a ;
!2
3
1221
−⋅⋅−
=
2
3
()( )
=a ;
3
−⋅⋅−
!3
()( )
=a . X
5
5
1321
−=
5
4
6
1521
−⋅⋅−
!5
4
()( )
=a ;
9
120
!4
3
−=−=
40
7
1421
−⋅⋅−
=
24
Ɋɹɞ ɬɚɤɠɟ ɦɨɠɟɬ ɛɵɬɶ ɡɚɞɚɧ ɪɟɤɭɪɪɟɧɬɧɵɦ ɫɨɨɬɧɨɲɟɧɢɟɦ, ɫɜɹɡɵɜɚɸɳɢɦ
ɩɨɫɥɟɞɭɸɳɢɣ ɱɥɟɧ ɪɹɞɚ ɫ ɩɪɟɞɵɞɭɳɢɦ, ɩɪɢ ɷɬɨɦ ɡɚɞɚɸɬɫɹ ɨɞɢɧ ɢɥɢ ɧɟɫɤɨɥɶɤɨ
ɱɥɟɧɨɜ ɪɹɞɚ ɢ ɮɨɪɦɭɥɚ, ɩɨ ɤɨɬɨɪɨɣ ɧɚɯɨɞɹɬɫɹ ɩɨɫɥɟɞɭɸɳɢɟ ɱɥɟɧɵ ɪɹɞɚ.
ɉɪɢɦɟɪ 1.2.
Ɂɚɩɢɫɚɬɶ ɬɪɟɬɢɣ, ɱɟɬɜɟɪɬɵɣ ɢ ɩɹɬɵɣ ɱɥɟɧɵ ɪɹɞɚ, ɨɩɪɟɞɟɥɹɟɦɨɝɨ ɪɟɤɭɪɪɟɧɬɧɵɦɢ ɫɨɨɬɧɨɲɟɧɢɹɦɢ
,
11=a
,
32=a
.
32
⋅+⋅=
aaa
21
−−
nnn
Ɋɟɲɟɧɢɟ.
ɉɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɧɚɯɨɞɢɦ:
;
9133232
123
234
345
=⋅+⋅=⋅+⋅= aaa
;
27339232
=⋅+⋅=⋅+⋅= aaa
. X
819327232
=⋅+⋅=⋅+⋅= aaa
ȼɨɡɦɨɠɧɨ ɬɚɤɠɟ ɪɟɲɟɧɢɟ ɨɛɪɚɬɧɨɣ ɡɚɞɚɱɢ: ɩɨ ɢɡɜɟɫɬɧɵɦ ɧɟɫɤɨɥɶɤɢɦ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɦ ɱɥɟɧɚɦ ɪɹɞɚ ɡɚɩɢɫɚɬɶ ɮɨɪɦɭɥɭ ɟɝɨ ɨɛɳɟɝɨ ɱɥɟɧɚ.
ɉɪɢɦɟɪ 1.3.
Ɂɚɩɢɫɚɬɶ ɮɨɪɦɭɥɭ
9
4
1 ++++
1)
2)
3)
1 +−+−
6
2
1
1
3
1
1
1
11
6
3
n
-ɝɨ ɱɥɟɧɚ ɞɥɹ ɞɚɧɧɵɯ ɪɹɞɨɜ:
25
16
24
715
20
...
;
120
1
...
;
9
1
1
...
++++
37
.
9
Ɋɟɲɟɧɢɟ.
S
1. ɉɨ ɭɫɥɨɜɢɸ
ɳɢɣ ɱɥɟɧ ɪɹɞɚ ɢɦɟɟɬ ɜɢɞ
4
,1
2
n
=
a
n
9
,
6
.
!2n
16
,
==== aaaa
4321
24
25
=a
,
5
. Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɨɛ-
120
2. ɉɨ ɭɫɥɨɜɢɸ ɱɥɟɧɵ ɪɹɞɚ ɦɟɧɹɸɬ ɡɧɚɤ, ɧɚɱɢɧɚɹ ɫ «+», ɩɪɢ ɷɬɨɦ ɜ ɡɧɚɦɟɧɚ-
ɬɟɥɟ ɧɚɛɥɸɞɚɸɬɫɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɟ ɡɧɚɱɟɧɢɹ ɧɟɱɟɬɧɵɯ ɱɢɫɟɥ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
1
+
n
()
1
−
=
ɨɛɳɢɣ ɱɥɟɧ ɪɹɞɚ ɢɦɟɟɬ ɜɢɞ
3. ɉɨ ɭɫɥɨɜɢɸ
6422 2,112 3,202 4=+=+ =+ =+
a
n
23 4
1
. X
n
n
+=2
a
n
1
3
1
,
,
6
.
12
−
n
1
11
1
,
20
1
,
===== aaaaa
. Ɍɚɤ ɤɚɤ 3 = 2 +1,
54321
37
, ɬɨ ɨɛɳɢɣ ɱɥɟɧ ɪɹɞɚ ɢɦɟɟɬ ɜɢɞ
ȼ ɧɟɤɨɬɨɪɵɯ ɫɥɭɱɚɹɯ ɮɨɪɦɭɥɭ ɨɛɳɟɝɨ ɱɥɟɧɚ ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɢ ɞɥɹ ɪɹɞɚ,
ɡɚɞɚɧɧɨɝɨ ɪɟɤɭɪɪɟɧɬɧɵɦɢ ɫɨɨɬɧɨɲɟɧɢɹɦɢ. Ɍɚɤ, ɞɥɹ ɪɹɞɚ ɩɪɢɦɟɪɚ 1.2 ɧɟɫɥɨɠɧɨ
ɡɚɦɟɬɢɬɶ, ɱɬɨ ɮɨɪɦɭɥɚ ɟɝɨ ɨɛɳɟɝɨ ɱɥɟɧɚ
a
ɋɯɨɞɹɳɢɟɫɹ ɢ ɪɚɫɯɨɞɹɳɢɟɫɹ ɱɢɫɥɨɜɵɟ ɪɹɞɵ. Ʉɨɧɟɱɧɚɹ ɫɭɦɦɚ
ɱɥɟɧɨɜ ɪɹɞɚ (1.1) ɨɛɨɡɧɚɱɚɟɬɫɹ
ɢ ɧɚɡɵɜɚɟɬɫɹ ɟɝɨ
S
n
13−=n
n
.
n
-ɣ ɱɚɫɬɢɱɧɨɣ ɫɭɦɦɨɣ.
n
ɩɟɪɜɵɯ
ɑɚɫɬɢɱɧɵɟ ɫɭɦɦɵ
,
...,......,,,
aaaSaaSaS +++=+==
2121211 nn
ɨɛɪɚɡɭɸɬ ɱɢɫɥɨɜɭɸ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ
.
{}
S
n
ȿɫɥɢ ɫɭɳɟɫɬɜɭɟɬ ɤɨɧɟɱɧɵɣ ɩɪɟɞɟɥ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɱɚɫɬɢɱɧɵɯ ɫɭɦɦ
, ɬ. ɟ. ɜɵɩɨɥɧɹɟɬɫɹ ɪɚɜɟɧɫɬɜɨ
{}
S
n
lim
n
, (1.2)
n
∞→
SS
=
ɬɨ ɪɹɞ (1.1) ɧɚɡɵɜɚɟɬɫɹ
ɫɯɨɞɹɳɢɦɫɹ, ɚ ɡɧɚɱɟɧɢɟ ɩɪɟɞɟɥɚ – ɱɢɫɥɨ
ɧɚɡɵɜɚɟɬɫɹ
ɫɭɦɦɨɣ ɪɹɞɚ. ɋɢɦɜɨɥɢɱɟɫɤɢ ɫɯɨɞɹɳɢɣɫɹ ɪɹɞ ɦɨɠɟɬ ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧ ɪɚɜɟɧ-
ɫɬɜɨɦ:
∞
21
n
¦
n
=1
ɢɥɢ
Saaa
=++++ ......
.
Sa
=
n
ȿɫɥɢ ɩɪɟɞɟɥ (1.2) ɧɟ ɫɭɳɟɫɬɜɭɟɬ ɢɥɢ ɪɚɜɟɧ ɛɟɫɤɨɧɟɱɧɨɫɬɢ, ɬɨ ɪɹɞ (1.1) ɧɚɡɵɜɚɟɬɫɹ
ɪɚɫɯɨɞɹɳɢɦɫɹ.
10
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