Determine the power series for the following functions, with its radius of convergence?

Question

a.1/(1 –x + x2)

b. (4 + x)3/2

c. arcsin(x)

Answer 1

a) =1/`[1-(x-x^2)] = 1 +(x-x^2)+(x-x^2)^2 +++++(x-x^2)^n++++
I(x-x^2I<1
x-x^2 >0 x(1-x)>0 00 always
If x>1 or x<0
x^2-x<1 so x^2-x-1<0 x= (1+-sqrt(5))/2 1/2(1-sqrt(5) 1/2(1-sqrt5) < x <0 and 1 Interval of convergence
1/2(1-sqrt5) < x < 1/2(1+sqrt5)
b) = 4^3/2( 1+x/4)^3/2
callx/4=z and 3/2=a
(1+z)^a = 1+az +a(a-1)z^2/2! +a(a-1)(a-2)z^3/3!++++
IzI<1 so Ix/4I<1 and IxI<4 r=4
c) arc sin x = x +1/2*3 x^3 +1*3/2*4*5 x^5+1*3*5/2*4*6*7 x^7+++
IxI<=1