64 Higher Engineering Mathematics
Now try the following exercise
Exercise 30 Further problems on the
binomial series
In problems 1 to 5 expand in ascending powers of x
as far as the term in x 3 , using the binomial theorem.
State in each case the limits of x for which the series
is valid.
1.
1
(1 − x)
[1 + x + x 2 + x 3 + · · ·, |x| < 1]
2.
1
(1 + x) 2
[1 − 2x + 3x 2 − 4x 3 + · · ·, |x| < 1]
3.
1
(2 + x) 3
⎡
⎣
1
8
1 −
3x
2
+
3x 2
2
−
5x 3
4
+ · · ·
|x| < 2
⎤
⎦
4.
√
2 + x
⎡
⎣
√
2
1 +
x
4
−
x 2
32
+
x 3
128
− · · ·
|x| < 2
⎤
⎦
5.
1
√
1 + 3x
⎡
⎢
⎢
⎣
1 −
3
2
x +
27
8
x 2 −
135
16
x 3 + · · ·
|x| <
1
3
⎤
⎥
⎥
⎦
6. Expand (2 + 3x)
−6 to three terms. For what
values of x is the expansion valid?
⎡
⎢
⎢
⎣
1
64
1 − 9x +
189
4
x 2
|x| <
2
3
⎤
⎥
⎥
⎦
7. When x is very small show that:
(a)
1
(1 − x) 2 √
(1 − x)
≈ 1 +
5
2
x
(b)
(1 − 2x)
(1 − 3x) 4 ≈ 1 + 10x
(c)
√
1 + 5x
3
√
1 − 2x
≈ 1 +
19
6
x
8. If x is very small such that x 2 and higher powers may be neglected, determine the power
series for
√
x + 4
3
√
8 − x
5
(1 + x) 3
4 −
31
15
x
9. Express the following as power series in
ascending powers of x as far as the term in
x 2 . State in each case the range of x for which
the series is valid.
(a)
1 − x
1 + x
(b)
(1 + x)
3
(1 − 3x) 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 problems involving the
binomial theorem
Binomial expansions may be used for numerical approximations, 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 0.96r or (1 − 0.04)r and 1.02h or
(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.
Now try the following exercise
Exercise 30 Further problems on the
binomial series
In problems 1 to 5 expand in ascending powers of x
as far as the term in x 3 , using the binomial theorem.
State in each case the limits of x for which the series
is valid.
1.
1
(1 − x)
[1 + x + x 2 + x 3 + · · ·, |x| < 1]
2.
1
(1 + x) 2
[1 − 2x + 3x 2 − 4x 3 + · · ·, |x| < 1]
3.
1
(2 + x) 3
⎡
⎣
1
8
1 −
3x
2
+
3x 2
2
−
5x 3
4
+ · · ·
|x| < 2
⎤
⎦
4.
√
2 + x
⎡
⎣
√
2
1 +
x
4
−
x 2
32
+
x 3
128
− · · ·
|x| < 2
⎤
⎦
5.
1
√
1 + 3x
⎡
⎢
⎢
⎣
1 −
3
2
x +
27
8
x 2 −
135
16
x 3 + · · ·
|x| <
1
3
⎤
⎥
⎥
⎦
6. Expand (2 + 3x)
−6 to three terms. For what
values of x is the expansion valid?
⎡
⎢
⎢
⎣
1
64
1 − 9x +
189
4
x 2
|x| <
2
3
⎤
⎥
⎥
⎦
7. When x is very small show that:
(a)
1
(1 − x) 2 √
(1 − x)
≈ 1 +
5
2
x
(b)
(1 − 2x)
(1 − 3x) 4 ≈ 1 + 10x
(c)
√
1 + 5x
3
√
1 − 2x
≈ 1 +
19
6
x
8. If x is very small such that x 2 and higher powers may be neglected, determine the power
series for
√
x + 4
3
√
8 − x
5
(1 + x) 3
4 −
31
15
x
9. Express the following as power series in
ascending powers of x as far as the term in
x 2 . State in each case the range of x for which
the series is valid.
(a)
1 − x
1 + x
(b)
(1 + x)
3
(1 − 3x) 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 problems involving the
binomial theorem
Binomial expansions may be used for numerical approximations, 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 0.96r or (1 − 0.04)r and 1.02h or
(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.