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§2. äÏÓÔÁÔÏÞÎÙÅ ÐÒÉÚÎÁËÉ ÓÈÏÄÉÍÏÓÔÉ ÐÏÌÏÖÉÔÅÌØÎÙÈ ÒÑÄÏ× 9
òÅÛÅÎÉÅ: ðÒÉÍÅÎÑÅÍ ×ÔÏÒÏÊ ÐÒÉÚÎÁË ÓÒÁ×ÎÅÎÉÑ.
lim
n→∞
a
n
b
n
= lim
n→∞
ln(1 +
1
n
2
)
1
n
2
= lim
n→∞
n
2
ln
1 +
1
n
2
=
= lim
n→∞
ln
1 +
1
n
2
n
2
= ln
"
lim
n→∞
1 +
1
n
2
n
2
#
= ln e = 1 6= 0
(×ÔÏÒÏÊ ÚÁÍÅÞÁÔÅÌØÎÙÊ ÐÒÅÄÅÌ). óÌÅÄÏ×ÁÔÅÌØÎÏ, ÉÓÓÌÅÄÕÅÍÙÊ ÒÑÄ ÔÁËÖÅ
ÓÈÏÄÉÔÓÑ.
2.3. ðÒÉÚÎÁË ÓÈÏÄÉÍÏÓÔÉ äÁÌÁÍÂÅÒÁ
åÓÌÉ ÄÌÑ ÒÑÄÁ Ó ÐÏÌÏÖÉÔÅÌØÎÙÍÉ ÞÌÅÎÁÍÉ
(4)
∞
X
n=1
a
n
= a
1
+ a
2
+ a
3
+ . . . + a
n
+ . . . , a
n
> 0,
ÓÕÝÅÓÔ×ÕÅÔ ÐÒÅÄÅÌ
lim
n→∞
a
n+1
a
n
= q,
ÔÏ
1) ÐÒÉ q < 1 ÒÑÄ (4) ÓÈÏÄÉÔÓÑ,
2) ÐÒÉ q > 1 ÉÌÉ q = +∞ ÒÑÄ (4) ÒÁÓÈÏÄÉÔÓÑ,
3) ÐÒÉ q = 1 Ï ÓÈÏÄÉÍÏÓÔÉ ÉÌÉ ÒÁÓÈÏÄÉÍÏÓÔÉ ÒÑÄÁ (4) ÎÉÞÅÇÏ ÓËÁÚÁÔØ
ÎÅÌØÚÑ É × ÜÔÏÍ ÓÌÕÞÁÅ ÎÁÄÏ ÐÒÉÍÅÎÉÔØ ÄÒÕÇÏÊ ÐÒÉÚÎÁË ÓÈÏÄÉÍÏÓÔÉ.
ðÒÉÍÅÒ 7. éÓÓÌÅÄÏ×ÁÔØ ÎÁ ÓÈÏÄÉÍÏÓÔØ ÒÑÄ
∞
X
n=1
1
n!
.
òÅÛÅÎÉÅ:
a
n
=
1
n!
, a
n+1
=
1
(n + 1)!
.
ðÒÉÍÅÎÉÍ ÐÒÉÚÎÁË äÁÌÁÍÂÅÒÁ.
q = lim
n→∞
1
(n + 1)!
:
1
n!
= lim
n→∞
n!
(n + 1)!
= lim
n→∞
1
n + 1
= 0.
ôÁË ËÁË q = 0 < 1, ÔÏ ÐÏ ÐÒÉÚÎÁËÕ äÁÌÁÍÂÅÒÁ ÉÓÓÌÅÄÕÅÍÙÊ ÒÑÄ ÓÈÏÄÉÔÓÑ.
ðÒÉÍÅÒ 8. éÓÓÌÅÄÏ×ÁÔØ ÎÁ ÓÈÏÄÉÍÏÓÔØ ÒÑÄ
∞
X
n=1
5
n
n(n + 1)
.
§2. äÏÓÔÁÔÏÞÎÙÅ ÐÒÉÚÎÁËÉ ÓÈÏÄÉÍÏÓÔÉ ÐÏÌÏÖÉÔÅÌØÎÙÈ ÒÑÄÏ×                         9

    òÅÛÅÎÉÅ: ðÒÉÍÅÎÑÅÍ ×ÔÏÒÏÊ ÐÒÉÚÎÁË ÓÒÁ×ÎÅÎÉÑ.
                ln(1 + n12 )
                                                
       an                                      1
   lim    = lim      1       = lim n2 ln 1 + 2 =
  n→∞ bn    n→∞                n→∞             n
                    n2
                                  n 2     "         n 2 #
                                 1                   1
                = lim ln 1 + 2          = ln lim 1 + 2        = ln e = 1 6= 0
                  n→∞            n            n→∞   n
(×ÔÏÒÏÊ ÚÁÍÅÞÁÔÅÌØÎÙÊ ÐÒÅÄÅÌ). óÌÅÄÏ×ÁÔÅÌØÎÏ, ÉÓÓÌÅÄÕÅÍÙÊ ÒÑÄ ÔÁËÖÅ
ÓÈÏÄÉÔÓÑ.

2.3. ðÒÉÚÎÁË ÓÈÏÄÉÍÏÓÔÉ äÁÌÁÍÂÅÒÁ

åÓÌÉ ÄÌÑ ÒÑÄÁ Ó ÐÏÌÏÖÉÔÅÌØÎÙÍÉ ÞÌÅÎÁÍÉ
             X∞
(4)              an = a 1 + a 2 + a 3 + . . . + a n + . . . ,   an > 0,
                 n=1
ÓÕÝÅÓÔ×ÕÅÔ ÐÒÅÄÅÌ
                                       an+1
                                     lim    = q,
                                    n→∞ an
ÔÏ
      1) ÐÒÉ q < 1 ÒÑÄ (4) ÓÈÏÄÉÔÓÑ,
      2) ÐÒÉ q > 1 ÉÌÉ q = +∞ ÒÑÄ (4) ÒÁÓÈÏÄÉÔÓÑ,
      3) ÐÒÉ q = 1 Ï ÓÈÏÄÉÍÏÓÔÉ ÉÌÉ ÒÁÓÈÏÄÉÍÏÓÔÉ ÒÑÄÁ (4) ÎÉÞÅÇÏ ÓËÁÚÁÔØ
         ÎÅÌØÚÑ É × ÜÔÏÍ ÓÌÕÞÁÅ ÎÁÄÏ ÐÒÉÍÅÎÉÔØ ÄÒÕÇÏÊ ÐÒÉÚÎÁË ÓÈÏÄÉÍÏÓÔÉ.
     ðÒÉÍÅÒ 7. éÓÓÌÅÄÏ×ÁÔØ ÎÁ ÓÈÏÄÉÍÏÓÔØ ÒÑÄ
                                      ∞
                                     X   1
                                            .
                                     n=1
                                         n!
     òÅÛÅÎÉÅ:
                           1               1
                              , an+1 =
                             an =                .
                           n!           (n + 1)!
ðÒÉÍÅÎÉÍ ÐÒÉÚÎÁË äÁÌÁÍÂÅÒÁ.
                    1      1           n!            1
         q = lim         :    = lim          = lim       = 0.
             n→∞ (n + 1)! n!    n→∞ (n + 1)!   n→∞ n + 1

ôÁË ËÁË q = 0 < 1, ÔÏ ÐÏ ÐÒÉÚÎÁËÕ äÁÌÁÍÂÅÒÁ ÉÓÓÌÅÄÕÅÍÙÊ ÒÑÄ ÓÈÏÄÉÔÓÑ.
  ðÒÉÍÅÒ 8. éÓÓÌÅÄÏ×ÁÔØ ÎÁ ÓÈÏÄÉÍÏÓÔØ ÒÑÄ
                                ∞
                              X       5n
                                            .
                               n=1
                                   n(n + 1)

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