Конспект лекций по математическому анализу. Шерстнев А.Н. - 228 стр.

pO USLOWI@ jwk (!)j = j P vj (!) , P vj (!)j  2M . s U^ETOM MONOTONNOS-
                       n+k           n
                        j =1       j =1
TI POSLEDOWATELXNOSTI uj (!) IMEEM
 j P u (!)v (!)j  2M (u (!) , u (!) + u (!) , u (!)
   p
        n+k    n+k                   n+1          n+2              n+2    n+3
 k=1
                          + : : : , un+p (!) + un+p(!)) = 2Mun+1 (!) (! 2 ):
tAK KAK uk =) 0, SOGLASNO 140.3 POLU^AEM TREBUEMOE.
   p. 2. w UKAZANNYH WY[E OBOZNA^ENIQH DLQ L@BOGO " > 0 SU]ESTWUET
N = N (") TAKOE, ^TO PRI n > N DLQ WSEH ! 2
          jwk (!)j = jvn+1(!) + : : : + vn+k (!)j < " (k = 1; 2; : : :):
(\TO SLEDUET IZ 140.3, PRIMENENNOGO K RQDU P v (!)). sLEDOWATELXNO, S
                                                   1
                                                               j
                                                        j =1
U^ETOM (2) I MONOTONNOSTI fuj (!)g:
      Xp
         un+k (!)vn+k (!)  "(un+1 (!) , un+p(!)) + "jun+p(!)j  3"M:
       k=1
sNOWA W SILU 140.3 POLU^AEM TREBUEMOE. >
    3. p R I M E R. iSSLEDUEM NA RAWNOMERNU@ SHODIMOSTX RQD

                           X1 sin nx
(3)                                   ( > 0):
                           n=1 n
w SLU^AE > 1 RQD SHODITSQ ABSOL@TNO I RAWNOMERNO (SM. 140.4). w SLU-
^AE  1 PRIMENIM PRIZNAK dIRIHLE, POLAGAQ vn (x) = sin nx;n un(x) =
n, (=) 0). oSTAETSQ ISSLEDOWATX NA OGRANI^ENNOSTX SUMMY j P v (x)j =            j
                                                                          j =1
j sin x + sin 2x + : : : + sin nxj. s U^ETOM TOVDESTWA 2 sin 
 sin                  =
cos( , ) , cos( + ) IMEEM
sin x + sin 2x + : : : + sin nx
   = (2 sin x2 ),1 [2 sin x2 
 sin x + : : : + 2 sin x2 
 sin nx]
   = (2 sin x2 ),1 [cos x2 , cos 32x + cos 32x , : : : , cos(n + 21 )x]
   = (2 sin x2 ),1 [cos x2 , cos(n + 21 )x]; x 6= 2k (k 2 Z).
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