Ряды. - 10 стр.

UptoLike

ВУЗ: 

МГТУ ГА | Москва

Рубрика: 

10 §2. äÏÓÔÁÔÏÞÎÙÅ ÐÒÉÚÎÁËÉ ÓÈÏÄÉÍÏÓÔÉ ÐÏÌÏÖÉÔÅÌØÎÙÈ ÒÑÄÏ×
òÅÛÅÎÉÅ: ðÒÉÍÅÎÉÍ ÐÒÉÚÎÁË äÁÌÁÍÂÅÒÁ.
a
n
=
5
n
n(n + 1)
, a
n+1
=
5
n+1
(n + 1)(n + 2)
,
q = lim
n→∞
a
n+1
a
n
= lim
n→∞
5
n+1
n(n + 1)
(n + 1)(n + 2)5
n
= lim
n→∞
5n
n + 2
= 5.
ôÁË ËÁË q = 5 > 1, ÔÏ ÐÏ ÐÒÉÚÎÁËÕ äÁÌÁÍÂÅÒÁ ÉÓÓÌÅÄÕÅÍÙÊ ÒÑÄ ÒÁÓÈÏÄÉÔÓÑ.
ðÒÉÍÅÒ 9. éÓÓÌÅÄÏ×ÁÔØ ÎÁ ÓÈÏÄÉÍÏÓÔØ ÒÑÄ
1 +
2
2
2!
+
3
3
3!
+ . . . +
n
n
n!
+ . . .
òÅÛÅÎÉÅ: ðÒÉÍÅÎÉÍ ÐÒÉÚÎÁË äÁÌÁÍÂÅÒÁ.
a
n
=
n
n
n!
, a
n+1
=
(n + 1)
n+1
(n + 1)!
,
q = lim
n→∞
a
n+1
a
n
=
(n + 1)
n+1
n!
(n + 1)!n
n
=
(n + 1)
n
(n + 1)n!
n!(n + 1)n
n
=
= lim
n→∞
(n + 1)
n
n
n
= lim
n→∞
n + 1
n
n
= lim
n→∞
1 +
1
n
n
= e.
ôÁË ËÁË q = e > 1, ÔÏ ÐÏ ÐÒÉÚÎÁËÕ äÁÌÁÍÂÅÒÁ ÉÓÓÌÅÄÕÅÍÙÊ ÒÑÄ ÒÁÓÈÏÄÉÔÓÑ.
2.4. ðÒÉÚÎÁË ÓÈÏÄÉÍÏÓÔÉ ëÏÛÉ
åÓÌÉ ÄÌÑ ÒÑÄÁ Ó ÎÅÏÔÒÉÃÁÔÅÌØÎÙÍÉ ÞÌÅÎÁÍÉ
(5)
∞
X
n=1
a
n
= a
1
+ a
2
+ a
3
+ . . . + a
n
+ . . . , a
n
> 0,
ÓÕÝÅÓÔ×ÕÅÔ ÐÒÅÄÅÌ
lim
n→∞
n
√
a
n
= q,
ÔÏ
1) ÐÒÉ q < 1 ÒÑÄ (5) ÓÈÏÄÉÔÓÑ,
2) ÐÒÉ q > 1 ÉÌÉ q = +∞ ÒÑÄ (5) ÒÁÓÈÏÄÉÔÓÑ,
3) ÐÒÉ q = 1 Ï ÓÈÏÄÉÍÏÓÔÉ ÉÌÉ ÒÁÓÈÏÄÉÍÏÓÔÉ ÒÑÄÁ (5) ÎÉÞÅÇÏ ÓËÁÚÁÔØ
ÎÅÌØÚÑ É × ÜÔÏÍ ÓÌÕÞÁÅ ÎÁÄÏ ÐÒÉÍÅÎÉÔØ ÄÒÕÇÏÊ ÐÒÉÚÎÁË ÓÈÏÄÉÍÏÓÔÉ.
ðÒÉÍÅÒ 10. éÓÓÌÅÄÏ×ÁÔØ ÎÁ ÓÈÏÄÉÍÏÓÔØ ÒÑÄ
∞
X
n=1
1
(ln n)
n
.
10            §2. äÏÓÔÁÔÏÞÎÙÅ ÐÒÉÚÎÁËÉ ÓÈÏÄÉÍÏÓÔÉ ÐÏÌÏÖÉÔÅÌØÎÙÈ ÒÑÄÏ×

     òÅÛÅÎÉÅ: ðÒÉÍÅÎÉÍ ÐÒÉÚÎÁË äÁÌÁÍÂÅÒÁ.
                                5n                  5n+1
                      an =            ,      an+1 =          ,
                             n(n + 1)         (n + 1)(n + 2)
                   an+1              5n+1n(n + 1)           5n
            q = lim          = lim                  =   lim      = 5.
                n→∞ an         n→∞ (n + 1)(n + 2)5n    n→∞ n + 2

ôÁË ËÁË q = 5 > 1, ÔÏ ÐÏ ÐÒÉÚÎÁËÕ äÁÌÁÍÂÅÒÁ ÉÓÓÌÅÄÕÅÍÙÊ ÒÑÄ ÒÁÓÈÏÄÉÔÓÑ.
  ðÒÉÍÅÒ 9. éÓÓÌÅÄÏ×ÁÔØ ÎÁ ÓÈÏÄÉÍÏÓÔØ ÒÑÄ
                         22 33        nn
                             1+
                           + + ...+       + ...
                         2! 3!        n!
     òÅÛÅÎÉÅ: ðÒÉÍÅÎÉÍ ÐÒÉÚÎÁË äÁÌÁÍÂÅÒÁ.
                              nn           (n + 1)n+1
                        an = , an+1 =                 ,
                              n!            (n + 1)!
                 an+1   (n + 1)n+1n! (n + 1)n(n + 1)n!
         q = lim      =              =                    =
             n→∞ an      (n + 1)!nn       n!(n + 1)nn
                                               n              n
                        (n + 1)n
                                                         
                                          n+1                1
                 = lim           = lim             = lim 1 +      = e.
                    n→∞    nn       n→∞     n         n→∞    n
ôÁË ËÁË q = e > 1, ÔÏ ÐÏ ÐÒÉÚÎÁËÕ äÁÌÁÍÂÅÒÁ ÉÓÓÌÅÄÕÅÍÙÊ ÒÑÄ ÒÁÓÈÏÄÉÔÓÑ.

2.4. ðÒÉÚÎÁË ÓÈÏÄÉÍÏÓÔÉ ëÏÛÉ

åÓÌÉ ÄÌÑ ÒÑÄÁ Ó ÎÅÏÔÒÉÃÁÔÅÌØÎÙÍÉ ÞÌÅÎÁÍÉ
                ∞
                X
(5)                   an = a 1 + a 2 + a 3 + . . . + a n + . . . ,   an > 0,
                n=1
ÓÕÝÅÓÔ×ÕÅÔ ÐÒÅÄÅÌ
                                              √
                                       lim    n
                                                an = q,
                                       n→∞
ÔÏ
      1) ÐÒÉ q < 1 ÒÑÄ (5) ÓÈÏÄÉÔÓÑ,
      2) ÐÒÉ q > 1 ÉÌÉ q = +∞ ÒÑÄ (5) ÒÁÓÈÏÄÉÔÓÑ,
      3) ÐÒÉ q = 1 Ï ÓÈÏÄÉÍÏÓÔÉ ÉÌÉ ÒÁÓÈÏÄÉÍÏÓÔÉ ÒÑÄÁ (5) ÎÉÞÅÇÏ ÓËÁÚÁÔØ
         ÎÅÌØÚÑ É × ÜÔÏÍ ÓÌÕÞÁÅ ÎÁÄÏ ÐÒÉÍÅÎÉÔØ ÄÒÕÇÏÊ ÐÒÉÚÎÁË ÓÈÏÄÉÍÏÓÔÉ.
     ðÒÉÍÅÒ 10. éÓÓÌÅÄÏ×ÁÔØ ÎÁ ÓÈÏÄÉÍÏÓÔØ ÒÑÄ
                                        ∞
                                        X        1
                                                      .
                                        n=1
                                              (ln n)n

Страницы