Understanding Engineering Mathematics

This is all very well – but how do we actuallycalculatethis? Asngets larger and larger
it gets more and more difficult. In particular, what would happen ifnbecame infinitely
large – equivalently, interest is being added continuously. In this case the result can be
expressed as alimit:

£ lim

n→∞

C

(

1 +

I

100 n

)n

Puttingx=

I

100

and dropping the constant factorCthis shows that a limit of the form

lim

n→∞

(

1 +

x

n

)n

is very important. This limit is a function ofx(because thendisappears on taking the
limit), and is in fact a definition ofex:

ex=lim

n→∞

(

1 +

x

n

)n

( 4. 1 )

This may seem an unusual way to define a function, but it does show howexis intimately
related to an important process of accumulating compound interest and natural growth and
decay in general. In fact it can be shown thatexdefined in this way as a limit does indeed
satisfy all the laws of indices (see Section 4.2.4 below).
We therefore take the limit (Equation 4.1) as our definition of theexponential function
ex, which is also written exp(x). This definition can in fact be used to obtain the infinite
series forex. You can try this yourself, using the binomial theorem:

Problem 3

Show that

(

1 Y

x

n

)n

= 1 YxY

1

(

1 −

1

n

)

2!

x^2 Y

1

(

1 −

1

n

)(

1 −

2

n

)

3!

x^3 Y···

By the binomial theorem (106, 111
➤
):

(

1 +

x

n

)n

= 1 +n

x

n

+

n(n− 1 )

2!

(x

n

) 2

+

n(n− 1 )...(n− 2 )

3!

(x

n

) 3

+···

= 1 +x+

(

1 −

1

n

)

2!

x^2 +

1 ·

(

1 −

1

n

)(

1 −

2

n

)

3!

x^3 +···

Problem 4

Hence deduce thatexcan be represented by the infinite series

ex= 1 YxY

x^2

2!

Y

x^3

3!

Y···Y

xr

r!

Y···