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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

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