A.maths (20分)

呢題m&s我想說是年年都會出一題
而且我亦想講會考amaths亦是一至兩年都出一題
我建議你要練多d~~~~因為是很基礎的

1a(i)
(1+x/a)^r
=1+r(x/a)+r(r-1)(x/a)^2/2!+...
=1+(r/a)x+r(r-1)(x^2)/2a^2+...

Comparing the coefficient of x and x^2
r/a=2r/3----(1)
r(r-1)/2a^2=-1/18---(2)

From(2)
18r(r-1)=-2a^2
18r^2-18r=-2a^2
-9r^2+9r=a^2

From(1)
r/a=2r/3
r^2/a^2=4r^2/9
r^2/(-9r^2+9r)=4r^2/9
1/(-9r^2+9r)=4/9
9=-36r^2+36r
1=-4r^2+4r
4r^2-4r+1=0
(2r-1)^2=0
r=1/2

when r=1/2
(1/2)/a=2(1/2)/3
1/2a=1/3
2a=3
a=3/2

1a(ii)
-1<x/a<1
-a<x<a
-3/2<x<3/2

b)
(1-x/a)^r
=(1+x/(-a))^r
=(1+x/(-3/2))^(1/2
By the results of (a)
(1+x/a)^r
=1+r(x/a)+r(r-1)(x/a)^2/2!+...
=1+(r/a)x+r(r-1)(x^2)/2a^2+...
when a=-3/2, r=1/2
(1-x/a)^r
=1+[(1/2)/(-3/2)]x-(1/18)x^2+...
=1-x/3-(1/18)x^2+...

b(ii)
-3/2<x<3/2

As follows:

圖片參考：http://i707.photobucket.com/albums/ww74/stevieg90/03-83.jpg


圖片參考：http://i707.photobucket.com/albums/ww74/stevieg90/02-104.jpg


2010-07-14 16:13:55 補充：
http://i707.photobucket.com/albums/ww74/stevieg90/03-83.jpg
http://i707.photobucket.com/albums/ww74/stevieg90/02-104.jpg