Конспект лекций по математическому анализу. Шерстнев А.Н. - 100 стр.

zAME^AQ, ^TO j x1 +, tt j  jxj PRI jxj < 1, POLU^AEM
                       jrn (x)j  jxjn,1j ln(1 + x)j ! 0 (n ! 1):
   3.     u P R A V N E N I E. dOKAVITE RAWENSTWO 36.7.
   x59. iNTEGRALXNYJ PRIZNAK SHODIMOSTI ^ISLOWOGO RQDA
      pUSTX f (x) (x  0) NE WOZRASTAET. rQD P f (j ) SHODITSQ TTOGDA
                                                   1
   1.
                   Zx                             j =1
FUNKCIQ F (x) = 0 f (t) dt (x > 0) OGRANI^ENA.
  oTMETIM    SNA^ALA, ^TO F (x) OPREDELENA DLQ L@BOGO x > 0n (SM. 48.2).
pUSTX j=1 f (j ) SHODITSQ. tOGDA SU]ESTWUET K > 0 TAKOE, ^TO jP=1 f (j )  K
       P
       1

DLQ WSEH n. tAKIM OBRAZOM, DLQ L@BOGO x > 0
                    Zx         Z [x]+1          Z1 Z2       Z [x]+1
       F (x) = 0 f (t) dt  0 f (t) dt = 0 + 1 + : : : + [x]
                 f (0) + P f (j )  f (0) + K:
                           [x]
                              j =1
oBRATNO, ESLI F (x)  K (x  0), TO
  Xn         X
             n      Zj     Xn Zj               Zn
     f (j ) = f (j ) dt             f (t) dt = f (t) dt = F (n)  K: >
   j =1         j =1 j ,1       j ,1 j =1       0

      p R I M E R. rQD nP=1 n1p SHODITSQ, ESLI p > 1 I RASHODITSQ PRI p  1.
                        1
   2.
                             ( 1; ESLI 0  t  1,
fpRIMENITE P. 1 K f (t) = 1p ; ESLI t > 1. g
                               t
   x60. gEOMETRI^ESKIE PRILOVENIQ
   1. pLO]ADX KRIWOLINEJNOJ TRAPECII. aKKURATNOE OPREDELENIE PLO-
]ADI PLOSKOJ FIGURY BUDET DANO W RAZDELE \mERA vORDANA". pOKA MY
OBRA]AEMSQ K GEOMETRI^ESKOJ INTUICII.
   pUSTX f (x) (a  x  b) INTEGRIRUEMA PO rIMANU. w SOOTWETSTWII S
45.1 I OPREDELENIEM INTEGRALA rIMANA PLO]ADX FIGURY, ZAKL@^ENNOJ
                                            100