Конспект лекций по математическому анализу. Шерстнев А.Н. - 230 стр.

     pUSTX un(t) (a  t  b) NEPRERYWNY I RQD nP=1 un (t) SHODITSQ RAW-
                                                1
       3.
NOMERNO NA [a; b]. tOGDA
             Zx X  1           1 Zx
                               X
(1)              ( un(t)) dt =      un(t) dt (a  x  b);
               a               n=1a           n=1
PRI^EM RQD W PRAWOJ ^ASTI SHODITSQ RAWNOMERNO.
 w SILU P. 2 W DOKAZATELXSTWE NUVDAETSQ LI[X RAWNOMERNAQ SHODIMOSTX
RQDA W PRAWOJ ^ASTI (1). tREBUEMOE SLEDUET IZ OCENKI
         +p Z x
        nX                  Z x nX
                                 +p                        +p
                                                          nX
      j      a
                uk (t) dtj  j
                             a
                                    uk (t)j dt  (b , a)k     uk k[a;b]: >
            k=n+1                           k=n+1               k=n+1

       4.   pUSTX uj : [a; b] ! R | GLADKIE FUNKCII, PRI^EM
  (A) PRI NEKOTOROM c (a  c  b) SHODITSQ ^ISLOWOJ RQD P uj (c),
                                                        1
                                                                         j =1

  (B) RQD P u0j (x) SHODITSQ RAWNOMERNO NA [a; b].
          1
                 j =1

tOGDA RQD uj (x) SHODITSQ RAWNOMERNO NA [a; b] I ( P uj (x))0 = P u0j (x).
                   P1                               1             1
         j =1                                      j =1          j =1
 pOLOVIM
              X
              1
      '(x) = u0j (x) (a  x  b); vj (x) = uj (x) , uj (c) (j 2 N):
                        j =1
                 Zx           1 Zx
sOGLASNO P. 3 c '(t) dt = jP=1( c u0j (t) dt) = jP=1 vj (x), I RQD jP=1 vj (x) SHO-
                                                  1                  1

DITSQ RAWNOMERNO. pO\TOMU RAWNOMERNO SHODITSQ RQD jP=1 uj (x) =
                                                                     1

P
1
  [vj (x) + uj (c)]. pRI \TOM
j =1
                 X
                 1               X
                                 1            Zx                   X
                                                                   1
             [       uj (x)]0 = [ vj (x)]0 = [ c '(t) dt]0 = '(x) = u0j (x): >
              j =1                   j =1                         j =1


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