Understanding Engineering Mathematics

Problem 14.15

Obtain the Maclaurins series for the functions (i)ex (ii) cosx

(i)exis the easy one – it just keeps repeating on differentiation and we have

f(r)(x)=ex for allr

so

f(r)( 0 )=e^0 = 1

So the series is

ex= 1 +x+

x^2

2!

+

x^3

3!

+···

(ii) cosxis almost as easy:

f( 0 )=cos 0= 1

f′( 0 )=−sin 0= 0

f′′( 0 )=−cos 0=− 1

f′′′( 0 )=sin 0= 0

f(iv)( 0 )=cos 0= 1

and so on, giving the series

cosx= 1 −

x^2

2!

+

x^4

4!

−···

So far as the convergence of power series is concerned we can use tests such as the

ratio test described in Section 14.10. The results of such tests will usually depend on

xand so the series may only converge for certain (if any) values ofx. The values

ofxfor which the series converges is usually called theradius of convergence.For

example, the binomial series for( 1 −x)−^1 only converges for|x|<1. Clearly, if a

series does not converge for a particular value ofx, then it cannot represent a sensible

function at that value.

Problem 14.16

Investigate the convergence of

(i) ex= 1 YxY

x^2

2!

Y

x^3

3!

Y

x^4

4!

Y··· (ii) x−

x^2

2

Y

x^3

3

−

x^4

4

Y···

(i) For 1+x+

x^2

2!

+

x^3

3!

+

x^4

4!

+···we haveun=

xn

n!

and so

lim

n→∞

∣

∣

∣

∣

un+ 1

un

∣

∣

∣

∣=nlim→∞

∣

∣

∣

∣

xn+^1

(n+ 1 )!

n!

xn

∣

∣

∣

∣=nlim→∞

∣

∣

∣

∣

x

n+ 1

∣

∣

∣

∣=0 for all finitex