Higher Engineering Mathematics

70 NUMBER AND ALGEBRA

Problem 9. Develop a series for sinhxusing

Maclaurin’s series.

f(x)=sinhxf(0)=sinh 0=

e^0 −e−^0

2

= 0

f′(x)=coshxf′(0)=cosh 0=

e^0 +e−^0

2

= 1

f′′(x)=sinhxf′′(0)=sinh 0= 0

f′′′(x)=coshxf′′′(0)=cosh 0= 1

fiv(x)=sinhxfiv(0)=sinh 0= 0

fv(x)=coshxfv(0)=cosh 0= 1

Substituting in equation (5) gives:

sinhx=f(0)+xf′(0)+

x^2

2!

f′′(0)+

x^3

3!

f′′′(0)

+

x^4

4!

fiv(0)+

x^5

5!

fv(0)+···

= 0 +(x)(1)+

x^2

2!

(0)+

x^3

3!

(1)+

x^4

4!

(0)

+

x^5

5!

(1)+···

i.e.sinhx=x+

x^3

3!

+

x^5

5!

+···

(as obtained in Section 5.5)

Problem 10. Produce a power series for

cos^22 xas far as the term inx^6.

From double angle formulae, cos 2A=2 cos^2 A− 1
(see Chapter 18).

from which, cos^2 A=

1

2

(1+cos 2A)

and cos^22 x=

1

2

(1+cos 4x)

From Problem 1,

cosx= 1 −

x^2

2!

+

x^4

4!

−

x^6

6!

+···

hence cos 4x= 1 −

(4x)^2

2!

+

(4x)^4

4!

−

(4x)^6

6!

+···

= 1 − 8 x^2 +

32

3

x^4 −

256

45

x^6 +···

Thus cos^22 x=

1

2

(1+cos 4x)

=

1

2

(

1 + 1 − 8 x^2 +

32

3

x^4 −

256

45

x^6 +···

)

i.e. cos^22 x= 1 − 4 x^2 +

16

3

x^4 −

128

45

x^6 +···

Now try the following exercise.

Exercise 36 Further problems on

Maclaurin’s series

1. Determine the first four terms of the power
   series for sin 2xusing Maclaurin’s series.
   ⎡

⎢

⎣

sin 2x= 2 x−

4

3

x^3 +

4

15

x^5

−

8

315

x^7 +···

⎤

⎥

⎦

2. Use Maclaurin’s series to produce a power
   series for cosh 3xas far as the term inx^6.
   [
   1 +

9

2

x^2 +

27

8

x^4 +

81

80

x^6

]

3. Use Maclaurin’s theorem to determine the
   first three terms of the power series for

ln (1+ex).

[

ln 2+

x

2

+

x^2

8

]

4. Determine the power series for cos 4tas far
   as the term int^6.
   [
   1 − 8 t^2 +

32

3

t^4 −

256

45

t^6

]

5. Expand e

3

2 xin a power series as far as the

term inx^3.

[

1 +

3

2

x+

9

8

x^2 +

9

16

x^3

]

6. Develop, as far as the term inx^4 , the power

series for sec 2x.

[

1 + 2 x^2 +

10

3

x^4

]

7. Expand e^2 θcos 3θas far as the term inθ^2

using Maclaurin’s series.

[

1 + 2 θ−

5

2

θ^2

]

8. Determine the first three terms of the series
   for sin^2 xby applying Maclaurin’s theorem.
   [
   x^2 −

1

3

x^4 +

2

45

x^6 ···

]