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Сходимость числовых рядов в примерах и задачах. Учебное пособие

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   
ɌɗɁɚɯɚɪɨɜɚ
       
ɋɏɈȾɂɆɈɋɌɖɑɂɋɅɈȼɕɏɊəȾɈȼ
ȼɉɊɂɆȿɊȺɏɂɁȺȾȺɑȺɏ
 
ɍɱɟɛɧɨɟɩɨɫɨɛɢɟ
                    
Ɇɨɫɤɜɚȼɨɥɨɝɞɚ
©ɂɧɮɪɚɂɧɠɟɧɟɪɢɹª

ɍȾɄ ȻȻɄ
Ɂ
Ɋɟɰɟɧɡɟɧɬɵ
ɤɚɧɞɮɢɡɦɚɬɧɚɭɤɞɨɰȾ. Ⱥ. ɉɪɨɤɭɞɢɧ
ɞɨɰɆ. ɘ. Ƚɚɥɤɢɧɚ
          ɁɚɯɚɪɨɜɚɌɗ
Ɂ  ɋɯɨɞɢɦɨɫɬɶɱɢɫɥɨɜɵɯɪɹɞɨɜɜɩɪɢɦɟɪɚɯɢɡɚɞɚɱɚɯɭɱɟɛɧɨɟɩɨɫɨɛɢɟ
ɌɗɁɚɯɚɪɨɜɚ ± Ɇɨɫɤɜɚ  ȼɨɥɨɝɞɚ  ɂɧɮɪɚɂɧɠɟɧɟɪɢɹ  ±  ɫ  ɢɥ
,6%1
Ɋɚɫɫɦɚɬɪɢɜɚɸɬɫɹ ɨɛɳɢɟ ɩɨɧɹɬɢɹ ɱɢɫɥɨɜɵɯ ɪɹɞɨɜ ɢɯ ɫɜɨɣɫɬɜɚ ɫɯɨɞɢɦɨɫɬɶ ɩɨ ɨɩɪɟɞɟɥɟɧɢɸɧɟɨɛɯɨɞɢɦɵɣɩɪɢɡɧɚɤɫɯɨɞɢɦɨɫɬɢɚɬɚɤɠɟɜɫɟɨɫɧɨɜɧɵɟɩɪɢɡɧɚɤɢɫɯɨ ɞɢɦɨɫɬɢ ɱɢɫɥɨɜɵɯ ɪɹɞɨɜ ɫ ɧɟɨɬɪɢɰɚɬɟɥɶɧɵɦɢ ɱɥɟɧɚɦɢ ɉɨɫɨɛɢɟ ɫɨɞɟɪɠɢɬ ɤɪɚɬɤɭɸ ɬɟɨɪɢɸ ɛɨɥɶɲɨɟ ɤɨɥɢɱɟɫɬɜɨ ɩɪɢɦɟɪɨɜɫ ɪɟɲɟɧɢɹɦɢ ɢɩɨɹɫɧɟɧɢɹɦɢ ɚ ɬɚɤɠɟ ɡɚɞɚɱɢ ɞɥɹɫɚɦɨɫɬɨɹɬɟɥɶɧɨɝɨɪɟɲɟɧɢɹɩɨɤɚɠɞɨɣɬɟɦɟɫɨɬɜɟɬɚɦɢɢɭɤɚɡɚɧɢɹɦɢ
Ⱦɥɹɢɫɩɨɥɶɡɨɜɚɧɢɹɜɭɱɟɛɧɨɦɩɪɨɰɟɫɫɟɜɬɟɯɧɢɱɟɫɤɢɯɜɵɫɲɢɯɭɱɟɛɧɵɯɡɚɜɟɞɟɧɢ ɹɯɩɪɢɢɡɭɱɟɧɢɢɬɟɦɵ©ɑɢɫɥɨɜɵɟɪɹɞɵªɪɚɡɞɟɥɚ©Ɋɹɞɵªɞɢɫɰɢɩɥɢɧ©Ɇɚɬɟɦɚɬɢɱɟɫɤɢɣ ɚɧɚɥɢɡªɢ©ȼɵɫɲɚɹɦɚɬɟɦɚɬɢɤɚª
  
,6%1 ɁɚɯɚɪɨɜɚɌɗ
ɂɡɞɚɬɟɥɶɫɬɜɨ©ɂɧɮɪɚɂɧɠɟɧɟɪɢɹª  Ɉɮɨɪɦɥɟɧɢɟɂɡɞɚɬɟɥɶɫɬɜɨ©ɂɧɮɪɚɂɧɠɟɧɟɪɢɹª
 ɍȾɄ ȻȻɄ

ɋɈȾȿɊɀȺɇɂȿ
ȼɜɟɞɟɧɢɟ
ɉɨɧɹɬɢɟɱɢɫɥɨɜɨɝɨɪɹɞɚɢɟɝɨɫɯɨɞɢɦɨɫɬɶ
ɇɟɨɛɯɨɞɢɦɨɟɭɫɥɨɜɢɟɫɯɨɞɢɦɨɫɬɢɪɹɞɚ
ɋɜɨɣɫɬɜɚɱɢɫɥɨɜɵɯɪɹɞɨɜ
ɉɪɢɡɧɚɤȾɚɥɚɦɛɟɪɚ
ɉɪɢɡɧɚɤɊɚɚɛɟ
ɊɚɞɢɤɚɥɶɧɵɣɩɪɢɡɧɚɤɄɨɲɢ
ɂɧɬɟɝɪɚɥɶɧɵɣɩɪɢɡɧɚɤɫɯɨɞɢɦɨɫɬɢɪɹɞɚ
ɉɪɢɡɧɚɤɫɪɚɜɧɟɧɢɹ
ɉɪɟɞɟɥɶɧɵɣɩɪɢɡɧɚɤɫɪɚɜɧɟɧɢɹ
Ɉɬɜɟɬɵɢɭɤɚɡɚɧɢɹɤɡɚɞɚɱɚɦɞɥɹɫɚɦɨɫɬɨɹɬɟɥɶɧɨɝɨɪɟɲɟɧɢɹ
Ʉɨɧɬɪɨɥɶɧɵɟɜɨɩɪɨɫɵɢɡɚɞɚɧɢɹɞɥɹɫɚɦɨɩɪɨɜɟɪɤɢ
Ɉɬɜɟɬɵɢɭɤɚɡɚɧɢɹɤɤɨɧɬɪɨɥɶɧɵɦɜɨɩɪɨɫɚɦɢɡɚɞɚɧɢɹɦ
ɞɥɹɫɚɦɨɩɪɨɜɟɪɤɢ
Ɂɚɞɚɧɢɹɞɥɹɩɪɨɜɟɪɨɱɧɵɯɪɚɛɨɬ
Ɂɚɤɥɸɱɟɧɢɟ
ɉɪɢɥɨɠɟɧɢɹ
ɋɩɢɫɨɤɥɢɬɟɪɚɬɭɪɵ

ȼɜɟɞɟɧɢɟ
ɍɱɟɛɧɨɟɩɨɫɨɛɢɟɪɚɫɫɦɚɬɪɢɜɚɟɬ ɬɟɦɵ ɪɚɡɞɟɥɚ ©ɑɢɫɥɨɜɵɟɪɹɞɵªɤɭɪɫɚɦɚ
ɬɟɦɚɬɢɱɟɫɤɨɝɨɚɧɚɥɢɡɚɌɟɨɪɢɹɪɹɞɨɜɨɱɟɧɶɨɛɲɢɪɧɚɢɦɧɨɝɨɝɪɚɧɧɚɂɫɫɥɟɞɨɜɚ
ɧɢɟ ɫɯɨɞɢɦɨɫɬɢ ɱɢɫɥɨɜɵɯ ɪɹɞɨɜ ɹɜɥɹɟɬɫɹ ɦɢɧɢɦɚɥɶɧɵɦ ɢ ɧɟɨɛɯɨɞɢɦɵɦ ɧɚɜɵ
ɤɨɦɞɥɹɩɨɧɢɦɚɧɢɹɫɭɬɢɪɹɞɨɜɢɜɨɡɦɨɠɧɨɫɬɢɢɯɩɪɚɤɬɢɱɟɫɤɨɝɨɩɪɢɦɟɧɟɧɢɹ
ɐɟɥɶ ɩɨɫɨɛɢɹ ± ɧɚ ɛɨɥɶɲɨɦ ɤɨɥɢɱɟɫɬɜɟ ɪɚɡɥɢɱɧɵɯ ɩɪɢɦɟɪɨɜ ɩɪɨɞɟɦɨɧ
ɫɬɪɢɪɨɜɚɬɶɨɛɳɢɟ ɩɨɧɹɬɢɹ ɪɹɞɨɜɢɢɯ ɫɯɨɞɢɦɨɫɬɢ ɚɬɚɤɠɟɨɯɜɚɬɢɬɶ ɩɪɢɡɧɚɤɢ
ɫɯɨɞɢɦɨɫɬɢɪɹɞɨɜɫɧɟɨɬɪɢɰɚɬɟɥɶɧɵɦɢɱɥɟɧɚɦɢɉɨɷɬɨɦɭɩɟɪɜɚɹɱɚɫɬɶɩɨɫɨɛɢɹ
ɤɚɫɚɟɬɫɹ ɨɛɳɢɯ ɩɨɧɹɬɢɣ ɱɢɫɥɨɜɵɯ ɪɹɞɨɜ ɢɯ ɫɜɨɣɫɬɜ ɩɪɢɡɧɚɤɨɜ ɫɯɨɞɢɦɨɫɬɢ
ɪɚɛɨɬɟ ɫ ɨɩɪɟɞɟɥɟɧɢɹɦɢ ȼɨ ɜɬɨɪɨɣ ɱɚɫɬɢ ɪɚɫɫɦɚɬɪɢɜɚɸɬɫɹ ɱɢɫɥɨɜɵɟ ɪɹɞɵ ɫ
ɧɟɨɬɪɢɰɚɬɟɥɶɧɵɦɢ ɱɥɟɧɚɦɢ ɩɪɢɡɧɚɤɢ ɢɯ ɫɯɨɞɢɦɨɫɬɢ ɢ ɨɛɥɚɫɬɶ ɩɪɢɦɟɧɟɧɢɹ
Ɂɚɤɥɸɱɢɬɟɥɶɧɚɹɱɚɫɬɶɫɨɫɬɨɢɬɢɡɤɨɧɬɪɨɥɶɧɵɯɜɨɩɪɨɫɨɜɜɬɟɫɬɨɜɨɣɮɨɪɦɟɢɡɚ
ɞɚɧɢɣɩɨɡɜɨɥɹɸɳɢɯɩɪɨɜɟɪɢɬɶɨɫɜɨɟɧɢɟɦɚɬɟɪɢɚɥɚɌɚɤɠɟɩɪɢɜɨɞɹɬɫɹɧɟɤɨɬɨ
ɪɵɟ ɜɚɪɢɚɧɬɵ ɤɨɬɨɪɵɟ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɧɚ ɫɚɦɨɫɬɨɹɬɟɥɶɧɵɯ ɢ ɤɨɧɬɪɨɥɶ
ɧɵɯɪɚɛɨɬɚɯɩɨɢɡɭɱɟɧɧɵɦɬɟɦɚɦ
ȼɟɫɶɬɟɨɪɟɬɢɱɟɫɤɢɣɦɚɬɟɪɢɚɥ ɫɨɩɪɨɜɨɠɞɚɟɬɫɹ ɛɨɥɶɲɢɦɤɨɥɢɱɟɫɬɜɨɦ ɩɪɢ
ɦɟɪɨɜɫɩɨɞɪɨɛɧɵɦɪɟɲɟɧɢɟɦɨɯɜɚɬɵɜɚɸɳɢɯɪɚɡɧɵɟɫɥɭɱɚɢɫɤɨɬɨɪɵɦɢɦɨɝɭɬ
ɜɫɬɪɟɬɢɬɶɫɹ ɫɬɭɞɟɧɬɵ ɩɪɢ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɣ ɪɚɛɨɬɟ ɢɡɭɱɚɹ ©Ɍɟɨɪɢɸɪɹɞɨɜª
Ʉɪɨɦɟ ɬɨɝɨ ɜ ɤɚɠɞɨɦ ɪɚɡɞɟɥɟ ɩɪɢɫɭɬɫɬɜɭɸɬ ɡɚɞɚɧɢɹ ɩɨɡɜɨɥɹɸɳɢɟ ɨɬɪɚɛɨɬɚɬɶ
ɢɡɥɨɠɟɧɧɵɟɦɟɬɨɞɵɄɨɜɫɟɦɡɚɞɚɧɢɹɦɩɪɢɜɟɞɟɧɵɨɬɜɟɬɵɤɧɟɤɨɬɨɪɵɦ±ɩɪɢɜɟ
ɞɟɧɵɭɤɚɡɚɧɢɹɤɪɟɲɟɧɢɸɢɩɨɹɫɧɟɧɢɹ
ɋɬɪɭɤɬɭɪɚ ɩɨɫɨɛɢɹ ɩɨɡɜɨɥɹɟɬ ɡɚɢɧɬɟɪɟɫɨɜɚɧɧɨɦɭ ɱɢɬɚɬɟɥɸ ɫɚɦɨɫɬɨɹ
ɬɟɥɶɧɨ ɨɫɜɚɢɜɚɬɶ ɤɨɧɬɪɨɥɢɪɨɜɚɬɶ ɢ ɨɰɟɧɢɜɚɬɶ ɫɬɟɩɟɧɶ ɩɨɧɢɦɚɧɢɹ ɦɚɬɟɪɢɚɥɚ
ɞɚɟɬɜɨɡɦɨɠɧɨɫɬɶɩɪɢɧɟɨɛɯɨɞɢɦɨɫɬɢɜɟɪɧɭɬɶɫɹɤɬɟɦɟɞɥɹɟɺɥɭɱɲɟɝɨɩɨɧɢ
ɦɚɧɢɹɩɨɷɬɨɦɭ ɞɚɧɧɨɟ ɩɨɫɨɛɢɟɹɜɥɹɟɬɫɹ ɭɞɨɛɧɵɦ ɢɩɨɥɟɡɧɵɦɞɨɩɨɥɧɟɧɢɟɦ ɤ
ɥɟɤɰɢɨɧɧɨɦɭɦɚɬɟɪɢɚɥɭɞɥɹɚɭɞɢɬɨɪɧɵɯɡɚɧɹɬɢɣɢɜɧɟɚɭɞɢɬɨɪɧɨɣɪɚɛɨɬɵ
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ɉɨɧɹɬɢɟɱɢɫɥɨɜɨɝɨɪɹɞɚɢɟɝɨɫɯɨɞɢɦɨɫɬɶ
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Ɉɩɪɟɞɟɥɟɧɢɟ ɋɭɦɦɚn ɩɟɪɜɵɯɱɥɟɧɨɜɪɹɞɚɧɚɡɵɜɚɟɬɫɹ n-ɨɣ ɱɚɫɬɢɱɧɨɣ
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Ɉɩɪɟɞɟɥɟɧɢɟ ɑɢɫɥɨɜɨɣɪɹɞ
ɞɨɜɚɬɟɥɶɧɨɫɬɶ ɟɝɨ ɱɚɫɬɢɱɧɵɯ ɫɭɦɦ
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ɧɟɱɧɵɣɢɥɢɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶɱɚɫɬɢɱɧɵɯɫɭɦɦ
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Ɉɬɜɟɬɪɹɞɫɯɨɞɢɬɫɹ
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ɊɟɲɟɧɢɟɊɹɞɪɚɫɯɨɞɢɬɫɹȾɨɤɚɠɟɦɷɬɨɩɨɨɩɪɟɞɟɥɟɧɢɸ
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ɪɚɫɯɨɞɢɬɫɹ
Ɉɬɜɟɬɪɹɞɪɚɫɯɨɞɢɬɫɹ
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ɉɪɢɦɟɪɂɫɫɥɟɞɨɜɚɬɶɫɯɨɞɢɦɨɫɬɶɪɹɞɚ
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Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɢ ɩɪɢ q ɪɹɞ ɪɚɫɯɨɞɢɬɫɹ ɬɨ ɟɫɬɶ ɪɹɞ
ɞɢɬɫɹɩɪɢ
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ɗɬɨɨɡɧɚɱɚɟɬ ɱɬɨ ɧɟɫɭɳɟɫɬɜɭɟɬ ɩɪɟɞɟɥɚ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢɱɚɫɬɢɱɧɵɯ
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ɉɪɢɦɟɪȾɨɤɚɡɚɬɶɱɬɨɪɹɞ
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ɬɨɩɨɨɩɪɟɞɟɥɟɧɢɸɢɦɟɟɦ
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ɫɥɟɞɭɸɳɢɦɨɛɪɚɡɨɦ
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S ɩɪɢ
n
fon 
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fofo
n
n
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nn
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n
fofo
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