Use the binomial theorem to expand square root of (4+x) in ascending powers of x to four terms. Give the limits of x for which your expansion is valid?

Question

5321 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-02-24T07:34:16

The series is #=2+x/4-x^2/64 +x^3/512+...# for #-4< x <4 #

The binomial theorem states

#(a+b)^n=a^n+n/1a^(n-1)b+(n(n-1))/(1*2)a^(n-2)b^2+(n(n-1)(n-2))/(1*2*3)a^(n-3)b^3+......#

Here, we have to rewrite

#sqrt(4+x)=2(1+x/4)^(1/2)#

#a=1#

#b=x/4#

#n=1/2#

Therefore,

#2(1+x/4)^(1/2)=2(1+(1/2)*(x/4)+((1/2)*(-1/2))/(1*2)(x/4)^2+((1/2)*(-1/2)*(-3/2))/(1*2*.3))(x/4)^3+.....#

#=2+x/4-x^2/64 +x^3/512+...#

This is valid for

#|x/4|<1#

That is, #-4 < x < 4 #