find the expansion of {( 1 + x^2) / √(1 + 4X)} i) in ascending powers of x upto and including the term x^2

Question

ii) state the values of x for which this expansion is valid

iii)Deduce the equation of the tangent to the curve { y = ( 1 + x^2) / (root of (1 + 4X)) } at a point where x = 0

this comes under the binomial expansions chapter

Answer 1

1/sqrt(1+4x)= (1+4x)^-1/2 = 1-1/2(4x) +(-1/2)(-3/2)(4x)^2/2+++
I4xI<1 so -1/4 so we get
(1+x^2) (1-2x +6x^2))= 1-2x+6x^2+x^2 =1-2x+7x^2+
This is the McLaurin expansion so
-2=f´(0) and
y-1=-2x
so2x+y-1=0 is the equation of the tangent at x=0

Answer 2

( 1 + x^2 ) / sqrt( 1 + 4x )
= (1+x^2) ( 1 - 2x ...) Expanding using Binomail theorem
= 1 - 2x + x^2 ,,,