Введение в математический анализ. Клевчихин Ю.А. - 15 стр.

UptoLike

ВУЗ: 

ДВФУ | Владивосток

Составители: 

Рубрика: 

n = 1
(x + y)
1
=
1
X
k=0
C
k
n
x
n−k
y
k
= C
0
1
x
1
y
0
+ C
1
1
x
0
y
1
= x + y.
n P (n) ⇒ P (n + 1)
P (n) = (x + y)
n
=
n
X
k=0
C
k
n
x
n−k
y
k
(x + y)
n+1
= (x + y)(x + y)
n
P (n)
= (x + y)
n
X
k=0
C
k
n
x
n−k
y
k
=
=
n
X
k=0
C
k
n
x
n+1−k
y
k
+
n
X
k=0
C
k
n
x
n−k
y
k+1
= ∗
k + 1 = k
0
k
0
n + 1
∗ =
n
X
k=0
C
k
n
x
n+1−k
y
k
+
n+1
X
k
0
=1
C
k
0
−1
n
x
n+1−k
0
y
k
0
= ∗
k
0
k
k = 0
k
0
= n + 1
∗ = x
n+1
+
n
X
k=1
C
k
n
x
n+1−k
y
k
+
n
X
k=1
C
k−1
n
x
n+1−k
y
k
+ y
n+1
= ∗
C
0
n+1
=
1 C
n+1
n+1
= 1 C
k
n
+ C
k−1
n
= C
k
n+1
∗ = x
n+1
+
n
X
k=1
(C
k
n
+ C
k−1
n
)x
n+1−k
y
k
+ y
n+1
=
= C
0
n+1
x
n+1
y
0
+
n
X
k=1
C
k
n+1
x
n+1−k
y
k
+ C
n+1
n+1
x
0
y
n+1
=
=
n
X
k=0
C
k
n+1
x
n+1−k
y
k
.
Ëåêöèÿ 2                                                                                              15


   Ä î ê à ç à ò å ë ü ñ ò â î. Ïðè n = 1 èìååì î÷åâèäíîå ðàâåíñòâî:
                           1
                           X
            (x + y)1 =           Cnk xn−k y k = C10 x1 y 0 + C11 x0 y 1 = x + y.
                           k=0

Äîêàæåì òåïåðü ïðè ëþáîì n âåðíîñòü èìïëèêàöèè P (n) ⇒ P (n + 1), ãäå
                                                     n
                                                     X
                         P (n) =  (x + y)n =              Cnk xn−k y k 
                                                     k=0

   Èìååì,
                                                  åñëè
                                                P (n)             n
                                                                  X
                 n+1                          n âåðíî
        (x + y)          = (x + y)(x + y)          = (x + y)            Cnk xn−k y k =
                                                                  k=0
                         n
                         X                        n
                                                  X
                     =         Cnk xn+1−k y k +         Cnk xn−k y k+1 = ∗
                         k=0                      k=0
Âî âòîðîé ñóììå çàìåíèì èíäåêñ ñóììèðîâàíèÿ ïî ôîðìóëå k + 1 = k 0 .
Òîãäà k 0 áóäåò èçìåíÿòüñÿ îò 1 äî n + 1:
                     n
                     X                         n+1
                                               X           0            0    0
               ∗=          Cnk xn+1−k y k +           Cnk −1 xn+1−k y k = ∗
                     k=0                      k0 =1

Òåïåðü îïÿòü âî âòîðîé ñóììå âìåñòî k 0 áóäåì ïèñàòü k (îò ýòîãî íè÷åãî
íå èçìåíèòñÿ!) è â ïåðâîé ñóììå âûäåëèì ïåðâîå ñëàãàåìîå (ïðè k = 0), à
âî âòîðîé  ïîñëåäíåå (ïðè k 0 = n + 1):
                         n
                         X                        n
                                                  X
       ∗ = xn+1 +              Cnk xn+1−k y k +         Cnk−1 xn+1−k y k + y n+1 = ∗
                         k=1                      k=1
                                                                0
Îáúåäèíèì òåïåðü äâå ñóììû â îäíó è âîñïîëüçóåìñÿ ðàâåíñòâàìè: Cn+1 =
    n+1        k    k−1   k
1, Cn+1 = 1 è Cn + Cn = Cn+1 :
               n
               X
  ∗ = xn+1 +       (Cnk + Cnk−1 )xn+1−k y k + y n+1 =
               k=1
                                     n
                                     X
               0                                              n+1 0 n+1
            = Cn+1 xn+1 y 0 +               k
                                           Cn+1 xn+1−k y k + Cn+1 x y   =
                                     k=1
                                                                            n
                                                                            X
                                                                                   k
                                                                     =            Cn+1 xn+1−k y k .
                                                                            k=0

Страницы