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4 §1. þÉÓÌÏ×ÙÅ ÒÑÄÙ. îÅÏÂÈÏÄÉÍÙÊ ÐÒÉÚÎÁË ÓÈÏÄÉÍÏÓÔÉ ÒÑÄÁ
ÐÏÓÌÅ ÞÅÔ×ÅÒÔÏÇÏ ÞÌÅÎÁ ÒÑÄÁ
∞
P
n=1
1
5
n−1
É ÒÁ×ÅÎ
∞
X
n=1
1
5
n+3
=
∞
X
n=5
1
5
n−1
=
∞
X
n=1
1
5
n−1
−
1 +
1
5
+
1
5
2
+
1
5
3
=
5
4
−
1 −
1
5
4
1 −
1
5
=
1
500
.
÷ ÐÒÉ×ÅÄÅÎÎÙÈ ÐÒÉÍÅÒÁÈ ÐÏÓÌÅÄÏ×ÁÔÅÌØÎÏÓÔØ {S
n
} ÞÁÓÔÉÞÎÙÈ ÓÕÍÍ ÒÑÄÁ
×ÙÞÉÓÌÑÌÁÓØ ÄÏÓÔÁÔÏÞÎÏ ÐÒÏÓÔÏ, ÔÁË ÞÔÏ ÓÕÝÅÓÔ×Ï×ÁÎÉÅ É ×ÅÌÉÞÉÎÁ ÐÒÅÄÅÌÁ
S
n
ÕÓÔÁÎÁ×ÌÉ×ÁÌÉÓØ ÎÅÐÏÓÒÅÄÓÔ×ÅÎÎÏ. ôÁËÉÍ ÏÂÒÁÚÏÍ, × ÓÉÌÕ ÏÐÒÅÄÅÌÅÎÉÑ
ÏÄÎÏ×ÒÅÍÅÎÎÏ ÄÏËÁÚÙ×ÁÌÁÓØ É ÓÈÏÄÉÍÏÓÔØ ÒÑÄÁ É ×ÙÞÉÓÌÑÌÁÓØ ÓÕÍÍÁ ÒÁÓ-
ÓÍÁÔÒÉ×ÁÅÍÏÇÏ ÒÑÄÁ. þÁÝÅ, ÏÄÎÁËÏ, ÎÅÐÒÏÓÒÅÄÓÔ×ÅÎÎÙÊ ÁÎÁÌÉÚ ÐÏÓÌÅÄÏ×Á-
ÔÅÌØÎÏÓÔÉ {S
n
} ÏÞÅÎØ ÓÌÏÖÅÎ, ÐÏÜÔÏÍÕ ÏÓÎÏ×ÎÏÊ ÚÁÄÁÞÅÊ × ÔÅÏÒÉÉ ÞÉÓÌÏ×ÙÈ
ÒÑÄÏ× Ñ×ÌÑÅÔÓÑ ÕÓÔÁÎÏ×ÌÅÎÉÅ ÓÈÏÄÉÍÏÓÔÉ ÉÌÉ ÒÁÓÈÏÄÉÍÏÓÔÉ ÄÁÎÎÏÇÏ ÒÑÄÁ
ÂÅÚ ×ÙÞÉÓÌÅÎÉÑ ÅÇÏ ÓÕÍÍÙ.
1.2. îÅÏÂÈÏÄÉÍÙÊ ÐÒÉÚÎÁË ÓÈÏÄÉÍÏÓÔÉ ÒÑÄÁ
åÓÌÉ ÒÑÄ
∞
X
n=1
a
n
= a
1
+ a
2
+ a
3
+ . . . + a
n
+ . . .
ÓÈÏÄÉÔÓÑ, ÔÏ ÅÇÏ ÏÂÝÉÊ ÞÌÅÎ a
n
ÓÔÒÅÍÉÔÓÑ Ë ÎÕÌÀ ÐÒÉ n → ∞, ÔÏ ÅÓÔØ
(4) lim
n→∞
a
n
= 0.
ðÒÉÍÅÒ 6. ðÒÏ×ÅÒÉÔØ, ×ÙÐÏÌÎÑÅÔÓÑ ÌÉ ÎÅÏÂÈÏÄÉÍÙÊ ÐÒÉÚÎÁË ÓÈÏÄÉÍÏ-
ÓÔÉ ÄÌÑ ÒÑÄÁ
1
2
+
2
3
+
3
4
+
4
5
+ . . .
òÅÛÅÎÉÅ: ïÂÝÉÊ ÞÌÅÎ a
n
=
n
n+1
. ôÁË ËÁË lim
n→∞
a
n
= lim
n→∞
n
n+1
= 1, ÔÏ
ÎÅÏÂÈÏÄÉÍÏÅ ÕÓÌÏ×ÉÅ (4) ÓÈÏÄÉÍÏÓÔÉ ÒÑÄÁ ÎÅ ×ÙÐÏÌÎÑÅÔÓÑ. óÌÅÄÏ×ÁÔÅÌØÎÏ,
ÄÁÎÎÙÊ ÒÑÄ ÒÁÓÈÏÄÉÔÓÑ.
ðÒÉÍÅÒ 7. ðÒÏ×ÅÒÉÔØ, ×ÙÐÏÌÎÑÅÔÓÑ ÌÉ ÎÅÏÂÈÏÄÉÍÏÅ ÕÓÌÏ×ÉÅ ÓÈÏÄÉÍÏÓÔÉ
ÄÌÑ ÒÑÄÁ
(5) 1 +
1
2
+
1
3
+
1
4
+
1
5
+ . . . +
1
n
+ . . .
òÅÛÅÎÉÅ: lim
n→∞
a
n
= lim
n→∞
1
n
= 0. úÎÁÞÉÔ ÎÅÏÂÈÏÄÉÍÏÅ ÕÓÌÏ×ÉÅ ÓÈÏÄÉÍÏÓÔÉ
ÒÑÄÁ ×ÙÐÏÌÎÅÎÏ.
4            §1. þÉÓÌÏ×ÙÅ ÒÑÄÙ. îÅÏÂÈÏÄÉÍÙÊ ÐÒÉÚÎÁË ÓÈÏÄÉÍÏÓÔÉ ÒÑÄÁ
                                  ∞
                                         1
                                  P
ÐÏÓÌÅ ÞÅÔ×ÅÒÔÏÇÏ ÞÌÅÎÁ ÒÑÄÁ           5n−1   É ÒÁ×ÅÎ
                                 n=1

 ∞           ∞           ∞                                           4
                                                            5 1 − 51
                                                       
X     1     X     1     X     1           1   1     1                      1
      n+3
          =       n−1
                      =       n−1
                                  −   1 +   +   2
                                                  +   3
                                                          =   −     1   =     .
n=1
    5       n=5
                5       n=1
                            5             5   5     5       4   1 − 5
                                                                          500

   ÷ ÐÒÉ×ÅÄÅÎÎÙÈ ÐÒÉÍÅÒÁÈ ÐÏÓÌÅÄÏ×ÁÔÅÌØÎÏÓÔØ {Sn } ÞÁÓÔÉÞÎÙÈ ÓÕÍÍ ÒÑÄÁ
×ÙÞÉÓÌÑÌÁÓØ ÄÏÓÔÁÔÏÞÎÏ ÐÒÏÓÔÏ, ÔÁË ÞÔÏ ÓÕÝÅÓÔ×Ï×ÁÎÉÅ É ×ÅÌÉÞÉÎÁ ÐÒÅÄÅÌÁ
Sn ÕÓÔÁÎÁ×ÌÉ×ÁÌÉÓØ ÎÅÐÏÓÒÅÄÓÔ×ÅÎÎÏ. ôÁËÉÍ ÏÂÒÁÚÏÍ, × ÓÉÌÕ ÏÐÒÅÄÅÌÅÎÉÑ
ÏÄÎÏ×ÒÅÍÅÎÎÏ ÄÏËÁÚÙ×ÁÌÁÓØ É ÓÈÏÄÉÍÏÓÔØ ÒÑÄÁ É ×ÙÞÉÓÌÑÌÁÓØ ÓÕÍÍÁ ÒÁÓ-
ÓÍÁÔÒÉ×ÁÅÍÏÇÏ ÒÑÄÁ. þÁÝÅ, ÏÄÎÁËÏ, ÎÅÐÒÏÓÒÅÄÓÔ×ÅÎÎÙÊ ÁÎÁÌÉÚ ÐÏÓÌÅÄÏ×Á-
ÔÅÌØÎÏÓÔÉ {Sn } ÏÞÅÎØ ÓÌÏÖÅÎ, ÐÏÜÔÏÍÕ ÏÓÎÏ×ÎÏÊ ÚÁÄÁÞÅÊ × ÔÅÏÒÉÉ ÞÉÓÌÏ×ÙÈ
ÒÑÄÏ× Ñ×ÌÑÅÔÓÑ ÕÓÔÁÎÏ×ÌÅÎÉÅ ÓÈÏÄÉÍÏÓÔÉ ÉÌÉ ÒÁÓÈÏÄÉÍÏÓÔÉ ÄÁÎÎÏÇÏ ÒÑÄÁ
ÂÅÚ ×ÙÞÉÓÌÅÎÉÑ ÅÇÏ ÓÕÍÍÙ.


1.2. îÅÏÂÈÏÄÉÍÙÊ ÐÒÉÚÎÁË ÓÈÏÄÉÍÏÓÔÉ ÒÑÄÁ

åÓÌÉ ÒÑÄ
                    ∞
                    X
                          an = a 1 + a 2 + a 3 + . . . + a n + . . .
                    n=1

ÓÈÏÄÉÔÓÑ, ÔÏ ÅÇÏ ÏÂÝÉÊ ÞÌÅÎ an ÓÔÒÅÍÉÔÓÑ Ë ÎÕÌÀ ÐÒÉ n → ∞, ÔÏ ÅÓÔØ
(4)                                    lim an = 0.
                                     n→∞

   ðÒÉÍÅÒ 6. ðÒÏ×ÅÒÉÔØ, ×ÙÐÏÌÎÑÅÔÓÑ ÌÉ ÎÅÏÂÈÏÄÉÍÙÊ ÐÒÉÚÎÁË ÓÈÏÄÉÍÏ-
ÓÔÉ ÄÌÑ ÒÑÄÁ
                         1 2 3 4
                          + + + + ...
                         2 3 4 5
                               n                         n
   òÅÛÅÎÉÅ: ïÂÝÉÊ ÞÌÅÎ an = n+1  . ôÁË ËÁË lim an = lim n+1 = 1, ÔÏ
                                                           n→∞         n→∞
ÎÅÏÂÈÏÄÉÍÏÅ ÕÓÌÏ×ÉÅ (4) ÓÈÏÄÉÍÏÓÔÉ ÒÑÄÁ ÎÅ ×ÙÐÏÌÎÑÅÔÓÑ. óÌÅÄÏ×ÁÔÅÌØÎÏ,
ÄÁÎÎÙÊ ÒÑÄ ÒÁÓÈÏÄÉÔÓÑ.
   ðÒÉÍÅÒ 7. ðÒÏ×ÅÒÉÔØ, ×ÙÐÏÌÎÑÅÔÓÑ ÌÉ ÎÅÏÂÈÏÄÉÍÏÅ ÕÓÌÏ×ÉÅ ÓÈÏÄÉÍÏÓÔÉ
ÄÌÑ ÒÑÄÁ
                            1 1 1 1      1
(5)                   1+     + + + + ...+ + ...
                            2 3 4 5      n
  òÅÛÅÎÉÅ: lim an = lim n1 = 0. úÎÁÞÉÔ ÎÅÏÂÈÏÄÉÍÏÅ ÕÓÌÏ×ÉÅ ÓÈÏÄÉÍÏÓÔÉ
             n→∞    n→∞
ÒÑÄÁ ×ÙÐÏÌÎÅÎÏ.

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