Конспект лекций по математическому анализу. Шерстнев А.Н. - 65 стр.

wELI^INA rn (x)  n1! f (n)(c)(x , a)n NAZYWAETSQ OSTATO^NYM ^LENOM W
FORME lAGRANVA.
  pOLOVIM DLQ a  z  x
                                 nX
                                  ,1 1
(2)       g(z) = f (x) , [f (z) + k! f (k) (z)(x , z)k + (x , z)n ]
                                 k=1
I WYBEREM  TAK, ^TOBY g(a) = 0. k FUNKCII g PRIMENIMA TEOREMA rOL-
LQ 32.1 (g(x) = 0 PO POSTROENI@, TAK ^TO g(a) = g(x)). sLEDOWATELXNO,
SU]ESTWUET c 2 (a; x) TAKOE, ^TO g0(c) = 0: pRQMOJ PODS^ET DAET
              g0(c) = , (n ,1 1)! f (n)(c)(x , c)n,1 + n(x , c)n,1;

OTKUDA  = n1! f (n)(c). tAK KAK g(a) = 0; IZ (2) POLU^AEM (1). >
   3. dLQ POLINOMA p(x) = a0 + a1x + : : : + an xn IMEEM rn+1 (x) = 0, TAK
^TO p(x) = p(a) + p0(a)(x , a) + : : : + n1! p(n) (a)(x , a)n. |TA FORMULA DAET,
W ^ASTNOSTI, RECEPT PREDSTAWLENIQ DANNOGO POLINOMA PO STEPENQM x , a.
   p R I M E R Y (ISPOLXZUETSQ FORMULA (1) PRI a = 0).
                       2                n,1      xn ex (x 2 R;  = (x) 2 (0; 1)).
   4. ex = 1 + x + x + : : : + x             +
                      2!            (n , 1)! n!
                     3        5                      2n,1
   5. sin x = x , x + x , : : : + (,1)n,1 x
                    3! 5!                         (2n , 1)!
                         x 2n+1
                   + (2n + 1)! sin(x + 2n 2+ 1 ) (x 2 R;  = (x) 2 (0; 1)).
                     2        4                       2n,2
   6. cos x = 1 , x + x , : : : + (,1)n,1 x
                    2! 4!                         (2n , 2)!
                                   x 2n
                                + (2n)! cos(x + n) (x 2 R;  = (x) 2 (0; 1)).
                            2
   7. ln(1 + x) = x , x + : : : + (,1)n x
                                                n,1
                                                     +  (,1)n+1xnn
                          2                   n , 1 n(1 + x)
                                                        (x > ,1;  = (x) 2 (0; 1)).
                                         1
                                      (n , 1)! b(b , 1) : : : (b , n + 2)x
   8. (1 + x)b = 1 + bx + : : : +                                          n,1

                + n1! b(b , 1) : : : (b , n + 1)xn (1 + x)b,n
                                                        (x > ,1;  = (x) 2 (0; 1)).

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