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òÑÄÙ
§1. þÉÓÌÏ×ÙÅ ÒÑÄÙ. îÅÏÂÈÏÄÉÍÙÊ ÐÒÉÚÎÁË ÓÈÏÄÉÍÏ-
ÓÔÉ ÒÑÄÁ
1.1. ïÂÝÉÅ ÐÏÎÑÔÉÑ
ðÕÓÔØ a
1
, a
2
, a
3
, . . . , a
n
, . . . ÐÒÏÉÚ×ÏÌØÎÙÅ ÞÉÓÌÁ. þÉÓÌÏ×ÙÍ ÒÑÄÏÍ ÎÁÚÙ×Á-
ÅÔÓÑ ×ÙÒÁÖÅÎÉÅ
(1)
∞
X
n=1
a
n
= a
1
+ a
2
+ a
3
+ . . . + a
n
+ . . . ,
ÞÉÓÌÁ a
1
, a
2
, a
3
, . . . , a
n
, . . . ÎÁÚÙ×ÁÀÔÓÑ ÞÌÅÎÁÍÉ ÒÑÄÁ, Á ×ÙÒÁÖÅÎÉÅ a
n
ËÁË
ÆÕÎËÃÉÑ ÞÉÓÌÁ n ÏÂÝÉÍ ÞÌÅÎÏÍ ÒÑÄÁ. åÓÌÉ ×ÍÅÓÔÏ n × ÆÏÒÍÕÌÕ ÏÂÝÅÇÏ
ÞÌÅÎÁ ÒÑÄÁ ÐÏÄÓÔÁ×ÌÑÔØ ÚÎÁÞÅÎÉÑ 1, 2, 3, . . . , ÔÏ ÍÏÖÎÏ ÎÁÊÔÉ ÓËÏÌØËÏ ÕÇÏÄÎÏ
ÞÌÅÎÏ× ÒÑÄÁ.
ðÒÉÍÅÒ 1. îÁÐÉÓÁÔØ ÞÅÔÙÒÅ ÐÅÒ×ÙÈ ÞÌÅÎÁ ÒÑÄÁ ÐÏ ÄÁÎÎÏÍÕ ÏÂÝÅÍÕ
ÞÌÅÎÕ a
n
=
1
n(n+1)
.
òÅÛÅÎÉÅ: ðÏÌÁÇÁÑ × ÆÏÒÍÕÌÅ ÏÂÝÅÇÏ ÞÌÅÎÁ ÐÏÓÌÅÄÏ×ÁÔÅÌØÎÏ ÚÎÁÞÅÎÉÑ
1, 2, 3, 4, ÐÏÌÕÞÉÍ:
a
1
=
1
1 · 2
=
1
2
, a
2
=
1
2 · 3
=
1
6
, a
3
=
1
3 · 4
=
1
12
, a
4
=
1
4 · 5
=
1
20
.
ðÒÉÍÅÒ 2. îÁÐÉÓÁÔØ ÆÏÒÍÕÌÕ ÏÂÝÅÇÏ ÞÌÅÎÁ ÄÌÑ ËÁÖÄÏÇÏ ÒÑÄÁ:
1) 1 +
1
2
+
1
3
+
1
4
+ . . .
2) 1 +
1
2
+
1
2
2
+
1
2
3
+
1
2
4
. . .
3)
2
5
+
4
8
+
6
11
+
8
14
. . .
òÅÛÅÎÉÅ: 1) úÎÁÍÅÎÁÔÅÌÉ ÞÌÅÎÏ× ÄÁÎÎÏÇÏ ÒÑÄÁ ÎÁÔÕÒÁÌØÎÙÊ ÒÑÄ ÞÉ-
ÓÅÌ. óÌÅÄÏ×ÁÔÅÌØÎÏ, ÏÂÝÉÊ ÞÌÅÎ a
n
=
1
n
.
2) úÎÁÍÅÎÁÔÅÌÉ ÞÌÅÎÏ× ÄÁÎÎÏÇÏ ÒÑÄÁ ÍÏÇÕÔ ÂÙÔØ ÐÏÌÕÞÅÎÙ ÉÚ ÆÏÒÍÕÌÙ
2
n−1
, ÇÄÅ n = 1, 2, 3, . . . óÌÅÄÏ×ÁÔÅÌØÎÏ, ÏÂÝÉÊ ÞÌÅÎ a
n
=
1
2
n−1
.
3) þÉÓÌÉÔÅÌÉ ÞÌÅÎÏ× ÄÁÎÎÏÇÏ ÒÑÄÁ ÞÅÔÎÙÅ ÞÉÓÌÁ ×ÉÄÁ 2n, Á ÚÎÁÍÅÎÁÔÅ-
ÌÉ ÞÉÓÌÁ, ËÏÔÏÒÙÅ ÍÏÇÕÔ ÂÙÔØ ÐÏÌÕÞÅÎÙ ÐÏ ÆÏÒÍÕÌÅ 3n+2. óÌÅÄÏ×ÁÔÅÌØÎÏ,
ÏÂÝÉÊ ÞÌÅÎ a
n
=
2n
3n+2
.
1
                                                  òÑÄÙ

    §1. þÉÓÌÏ×ÙÅ ÒÑÄÙ. îÅÏÂÈÏÄÉÍÙÊ ÐÒÉÚÎÁË ÓÈÏÄÉÍÏ-
     ÓÔÉ ÒÑÄÁ
1.1. ïÂÝÉÅ ÐÏÎÑÔÉÑ

ðÕÓÔØ a1 , a2 , a3, . . . , an , . . .         ÐÒÏÉÚ×ÏÌØÎÙÅ ÞÉÓÌÁ. þÉÓÌÏ×ÙÍ ÒÑÄÏÍ ÎÁÚÙ×Á-
ÅÔÓÑ ×ÙÒÁÖÅÎÉÅ
                              ∞
                              X
(1)                                 an = a 1 + a 2 + a 3 + . . . + a n + . . . ,
                              n=1

ÞÉÓÌÁ a1 , a2 , a3, . . . , an , . . . ÎÁÚÙ×ÁÀÔÓÑ ÞÌÅÎÁÍÉ ÒÑÄÁ, Á ×ÙÒÁÖÅÎÉÅ an ËÁË
ÆÕÎËÃÉÑ ÞÉÓÌÁ n ÏÂÝÉÍ ÞÌÅÎÏÍ ÒÑÄÁ. åÓÌÉ ×ÍÅÓÔÏ n × ÆÏÒÍÕÌÕ ÏÂÝÅÇÏ
ÞÌÅÎÁ ÒÑÄÁ ÐÏÄÓÔÁ×ÌÑÔØ ÚÎÁÞÅÎÉÑ 1, 2, 3, . . . , ÔÏ ÍÏÖÎÏ ÎÁÊÔÉ ÓËÏÌØËÏ ÕÇÏÄÎÏ
ÞÌÅÎÏ× ÒÑÄÁ.
    ðÒÉÍÅÒ 1. îÁÐÉÓÁÔØ ÞÅÔÙÒÅ ÐÅÒ×ÙÈ ÞÌÅÎÁ ÒÑÄÁ ÐÏ ÄÁÎÎÏÍÕ ÏÂÝÅÍÕ
                 1
ÞÌÅÎÕ an = n(n+1)     .
    òÅÛÅÎÉÅ: ðÏÌÁÇÁÑ × ÆÏÒÍÕÌÅ ÏÂÝÅÇÏ ÞÌÅÎÁ ÐÏÓÌÅÄÏ×ÁÔÅÌØÎÏ ÚÎÁÞÅÎÉÑ
1, 2, 3, 4, ÐÏÌÕÞÉÍ:
               1  1                       1  1                  1   1                   1   1
      a1 =       = ,             a2 =       = ,        a3 =        = ,         a4 =        = .
              1·2 2                      2·3 6                3 · 4 12                4 · 5 20
    ðÒÉÍÅÒ 2. îÁÐÉÓÁÔØ ÆÏÒÍÕÌÕ ÏÂÝÅÇÏ ÞÌÅÎÁ ÄÌÑ ËÁÖÄÏÇÏ ÒÑÄÁ:
     1) 1 + 21 + 13 + 14 + . . .
      2) 1 + 12 +    1
                     22
                          +   1
                              23
                                 + 214   ...
    3) 52 + 48 + +    6
                     11
                               8
                              14
                                 ...
   òÅÛÅÎÉÅ: 1) úÎÁÍÅÎÁÔÅÌÉ ÞÌÅÎÏ× ÄÁÎÎÏÇÏ ÒÑÄÁ ÎÁÔÕÒÁÌØÎÙÊ ÒÑÄ ÞÉ-
ÓÅÌ. óÌÅÄÏ×ÁÔÅÌØÎÏ, ÏÂÝÉÊ ÞÌÅÎ an = n1 .
   2) úÎÁÍÅÎÁÔÅÌÉ ÞÌÅÎÏ× ÄÁÎÎÏÇÏ ÒÑÄÁ ÍÏÇÕÔ ÂÙÔØ ÐÏÌÕÞÅÎÙ ÉÚ ÆÏÒÍÕÌÙ
 n−1                                                        1
2 , ÇÄÅ n = 1, 2, 3, . . . óÌÅÄÏ×ÁÔÅÌØÎÏ, ÏÂÝÉÊ ÞÌÅÎ an = 2n−1 .
   3) þÉÓÌÉÔÅÌÉ ÞÌÅÎÏ× ÄÁÎÎÏÇÏ ÒÑÄÁ ÞÅÔÎÙÅ ÞÉÓÌÁ ×ÉÄÁ 2n, Á ÚÎÁÍÅÎÁÔÅ-
ÌÉ ÞÉÓÌÁ, ËÏÔÏÒÙÅ ÍÏÇÕÔ ÂÙÔØ ÐÏÌÕÞÅÎÙ ÐÏ ÆÏÒÍÕÌÅ 3n+2. óÌÅÄÏ×ÁÔÅÌØÎÏ,
                   2n
ÏÂÝÉÊ ÞÌÅÎ an = 3n+2     .
                                       1

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