64 Higher Engineering Mathematics
Now try the following exercise
Exercise 30 Further problems on the
binomial series
In problems1 to5 expand inascending powers ofx
as far as the term inx^3 , using the binomial theorem.
Statein each casethelimits ofxfor which theseries
is valid.
1.
1
( 1 −x)
[1+x+x^2 +x^3 +···,|x|<1]
2.
1
( 1 +x)^2
[1− 2 x+ 3 x^2 − 4 x^3 +···,|x|<1]
3.
1
( 2 +x)^3
⎡
⎣
1
8
(
1 −
3 x
2
+
3 x^2
2
−
5 x^3
4
+···
)
|x|< 2
⎤
⎦
4.
√
2 +x
⎡
⎣
√
2
(
1 +
x
4
−
x^2
32
+
x^3
128
−···
)
|x|< 2
⎤
⎦
5.
1
√
1 + 3 x
⎡
⎢
⎢
⎣
(
1 −
3
2
x+
27
8
x^2 −
135
16
x^3 +···
)
|x|<
1
3
⎤
⎥
⎥
⎦
- Expand( 2 + 3 x)−^6 to three terms. For what
values ofxis the expansion valid?
⎡
⎢
⎢
⎣
1
64
(
1 − 9 x+
189
4
x^2
)
|x|<
2
3
⎤
⎥
⎥
⎦
- Whenxis very small show that:
(a)
1
( 1 −x)^2
√
( 1 −x)
≈ 1 +
5
2
x
(b)
( 1 − 2 x)
( 1 − 3 x)^4
≈ 1 + 10 x
(c)
√
1 + 5 x
√ (^31) − 2 x≈ 1 +
19
6
x
- Ifxis very small such thatx^2 and higher pow-
ers may be neglected, determine the power
series for
√
x+ 43
√
8 −x
√ (^5) ( 1 +x) 3
[
4 −
31
15
x
]
- Express the following as power series in
ascending powers ofxas far as the term in
x^2. State in each case the range ofxfor which
the series is valid.
(a)
√(
1 −x
1 +x
)
(b)
( 1 +x)^3
√
( 1 − 3 x)^2
√
( 1 +x^2 )
⎡
⎢
⎢
⎣
(a) 1−x+
1
2
x^2 ,|x|< 1
(b) 1−x−
7
2
x^2 ,|x|<
1
3
⎤
⎥
⎥
⎦
7.5 Practical problemsinvolving the
binomial theorem
Binomial expansionsmaybeusedfornumerical approx-
imations, for calculations with small variations and in
probability theory (see Chapter 57).
Problem 17. The radius of a cylinder is reduced
by 4% and its height is increased by 2%. Determine
the approximate percentage change in (a) its
volume and (b) its curved surface area, (neglecting
the products of small quantities).
Volume of cylinder=πr^2 h.
Let r and h be the original values of radius and
height.
The new values are 0.96ror( 1 − 0. 04 )rand 1.02hor
( 1 + 0. 02 )h.
(a) New volume=π[( 1 − 0. 04 )r]^2 [( 1 + 0. 02 )h]
=πr^2 h( 1 − 0. 04 )^2 ( 1 + 0. 02 )
Now( 1 − 0. 04 )^2 = 1 − 2 ( 0. 04 )+( 0. 04 )^2
=( 1 − 0. 08 ),
neglecting powers of small terms.