62 Higher Engineering Mathematics
- Expand(p+ 2 q)^11 as far as the fifth term.
⎡
⎣
p^11 + 22 p^10 q+ 220 p^9 q^2
+ 1320 p^8 q^3 + 5280 p^7 q^4
⎤
⎦
- Determine the sixth term of
(
3 p+
q
3
) 13
.
[34749p^8 q^5 ]
- Determine the middle term of( 2 a− 5 b)^8.
[700000a^4 b^4 ] - Use the binomial theorem to determine, cor-
rect to 4 decimal places:
(a)( 1. 003 )^8 (b)( 1. 042 )^7
[(a) 1.0243 (b) 1.3337] - Use the binomial theorem to determine, cor-
rect to 5 significant figures:
(a)( 0. 98 )^7 (b)( 2. 01 )^9
[(a) 0.86813 (b) 535.51] - Evaluate( 4. 044 )^6 correct to 3 decimal places.
[4373.880]
7.4 Further worked problems on the
binomial series
Problem 11.
(a) Expand
1
( 1 + 2 x)^3
in ascending powers ofxas
far as the term inx^3 , using the binomial series.
(b) State the limits ofxfor which the expansion
is valid.
(a) Using the binomial expansion of( 1 +x)n,where
n=−3andxis replaced by 2xgives:
1
( 1 + 2 x)^3
=( 1 + 2 x)−^3
= 1 +(− 3 )( 2 x)+
(− 3 )(− 4 )
2!
( 2 x)^2
+
(− 3 )(− 4 )(− 5 )
3!
( 2 x)^3 +···
= 1 − 6 x+ 24 x^2 − 80 x^3 +···
(b) The expansion is valid provided| 2 x|<1,
i.e. |x|<
1
2
or−
1
2
<x<
1
2
Problem 12.
(a) Expand
1
( 4 −x)^2
in ascending powers ofxas
far as the term inx^3 , using the binomial
theorem.
(b) What are the limits ofxfor which the expan-
sion in (a) is true?
(a)
1
( 4 −x)^2
=
1
[
4
(
1 −
x
4
)] 2 =
1
42
(
1 −
x
4
) 2
=
1
16
(
1 −
x
4
)− 2
Using the expansion of( 1 +x)n
1
( 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< 4
Problem 13.√ 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
)^12
Using the expansion of( 1 +x)n,
2
(
1 +
x
4
)^1
2
= 2
[
1 +
(
1
2
)(
x
4
)
+
( 1 / 2 )(− 1 / 2 )
2!
(x
4
) 2