Higher Engineering Mathematics, Sixth Edition

72 Higher Engineering Mathematics

Problem 11. Develop a series for sinhxusing

Maclaurin’s series.

f(x)=sinhxf( 0 )=sinh0=

e^0 −e−^0

2

= 0

f′(x)=coshxf′( 0 )=cosh0=

e^0 +e−^0

2

= 1

f′′(x)=sinhxf′′( 0 )=sinh0= 0

f′′′(x)=coshxf′′′( 0 )=cosh0= 1

fiv(x)=sinhxfiv( 0 )=sinh0= 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, page 49)

Problem 12. Produce a power series for cos^22 x

as far as the term inx^6.

From double angle formulae, cos2A=2cos^2 A−1(see

Chapter 17).

from which, cos^2 A=

1

2

( 1 +cos2A)

and cos^22 x=

1

2

( 1 +cos 4x)

From Problem 1,

cosx= 1 −

x^2

2!

+

x^4

4!

−

x^6

6!

+···

hence cos4x= 1 −

( 4 x)^2

2!

+

( 4 x)^4

4!

−

( 4 x)^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 32 Further problemson

Maclaurin’s series

1. Determine the first four terms of the power
   series for sin2x⎡using Maclaurin’s series.

⎢

⎣

sin2x= 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’stheorem to determine the first
   three terms of the power series for ln[ ( 1 +ex).
   ln2+

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 sec2x.

[

1 + 2 x^2 +

10

3

x^4

]

7. Expand e^2 θcos3θas far as the term inθ^2 using
   Maclaurin’s series.

[

1 + 2 θ−

5

2

θ^2

]

8. Determine thefirst threeterms of the series for
   sin^2 xby applying Maclaurin’s theorem.
   [
   x^2 −

1

3

x^4 +

2

45

x^6 ···

]

9. UseMaclaurin’s series to determinetheexpan-
   sion of( 3 + 2 t)^4.
   [
   81 + 216 t+ 216 t^2 + 96 t^3 + 16 t^4

]