62 NUMBER AND ALGEBRA= 1 +(−3)(2x)+(−3)(−4)
2!(2x)^2+(−3)(−4)(−5)
3!(2x)^3 +···= 1 − 6 x+ 24 x^2 − 80 x^3 +···(b) The expansion is valid provided| 2 x|<1,
i.e. |x|<1
2or−1
2<x<1
2Problem 11.(a) Expand1
(4−x)^2in ascending powers ofxas far as the term inx^3 , using the binomial
theorem.
(b) What are the limits of x for which the
expansion in (a) is true?(a)1
(4−x)^2=1
[
4(
1 −x
4)] 2 =142(
1 −x
4) 2=1
16(
1 −x
4)− 2Using the expansion of (1+x)n1
(4−x)^2=1
16(
1 −x
4)− 2=1
16[
1 +(−2)(
−x
4)+(−2)(−3)
2!(
−x
4) 2+(−2)(−3)(−4)
3!(
−x
4) 3
+···]=1
16(
1 +x
2+3 x^2
16+x^3
16+···)(b) The expansion in (a) is true provided∣
∣
∣x
4∣
∣
∣<1,i.e.|x|< 4 or− 4 <x< 4Problem 12. Use the binomial theorem to
expand√
4 +xin ascending powers ofxto
four terms. Give the limits ofxfor which the
expansion is valid.√
4 +x=√[
4(
1 +x
4)]=√
4√(
1 +x
4)
= 2(
1 +x
4)^1
2Using the expansion of (1+x)n,2(
1 +x
4)^1
2= 2[
1 +(
1
2)(
x
4)
+(1/2)(− 1 /2)
2!(x4) 2+(1/2)(− 1 /2)(− 3 /2)
3!(x4) 3
+···]= 2(
1 +x
8−x^2
128+x^3
1024−···)= 2 +x
4−x^2
64+x^3
512−···This is valid when∣
∣
∣x
4∣
∣
∣<1,i.e. |x|<4or− 4 <x< 4Problem 13. Expand1
√
(1− 2 t)in ascendingpowers oftas far as the term int^3.
State the limits oftfor which the expression is
valid.1
√
(1− 2 t)=(1− 2 t)−1
2= 1 +(
−1
2)
(− 2 t)+(− 1 /2)(− 3 /2)
2!(− 2 t)^2+(− 1 /2)(− 3 /2)(− 5 /2)
3!(− 2 t)^3 +···,using the expansion for (1+x)n= 1 +t+3
2t^2 +5
2t^3 +···The expression is valid when| 2 t|<1,i.e. |t|<1
2or −1
2<t<1
2