Loading [MathJax]/jax/output/HTML-CSS/jax.js

Binomial Theorem: nk=0(nk)xkynk=(x+y)n

nk=0(nk)xkynk=(x+y)n since:

(1)  For n=0,nk=0(nk)xkynk=(00)x0y0=1=(x+y)0

(2)   Assume that nk=0(nk)xkynk=(x+y)n 

(3)  (x+y)(x+y)n= =(x+y)(xn+(n1)xn1y+(n2)xn2y2++(nk)xnkyk+ +(nn1)xyn1+yn) 

=xn+1+[1+(n1)]xny+[(n1)+(n2)]xn1y2+ [(nk1)+(nk)]xnk+1yk++[(nn1)+1]xyn+yn+1

(4) Using Pascal's Rule (nr1)+(nr)=(n+1r) and 1=(n0)=(nn) so that:

xn+1+[1+(n1)]xny+[(n1)+(n2)]xn1y2+ [(nk1)+(nk)]xnk+1yk++[(nn1)+1]xyn+yn+1=

(n+10)xn+1+(n+11)xny+(n+12)xn1y2+ (n+1k)xnk+1yk++(n+1n1)xyn+(n+1n+1)yn+1=n+1k=0(n+1k)xkynk  

No comments:

Post a Comment

e: limn(1+1n)n = limnni=0(1i!)

limn(1+1n)n = limnni=0(1i!) since: (...