All we need here is to note that such things as 1/n, 2 /n,....‘tend to’ zero asn‘tends
to’ infinity, i.e. gets infinitely large. So:
ex=lim
n→∞(
1 +x
n)n=lim
n→∞
^1 +x+1(
1 −1
n)2!x^2 +1(
1 −1
n)(
1 −2
n)3!x^3 +···+1(
1 −1
n)
...(
1 −r− 1
n)r!xr+···
= 1 +x+x^2
2!+x^3
3!+···+xr
r!+···On letting all the terms with 1/n, 2 /n,...tend to zero. Hence we obtain the series form
forexfrom the limit definition:
ex= 1 +x+x^2
2!+x^3
3!+···+xr
r!+···=∑∞r= 0xr
r!In particular, we now have fore=e^1 :
e= 1 + 1 +1
2!+1
3!+···+1
r!+···We can calculate the value ofeto any required accuracy by taking a sufficient number
of terms of this series. For example, summing the first 11 terms of the above series (i.e.
r=10) giveseas 2.71828 to 5 decimal places. As noted earliereis anirrational number.
When your calculator gives you a ‘value’ fore, it is only an approximation to the available
number of decimal places – to give the exact value ofeit would need to have a display
of infinite length.
The graph of an exponential function is very simple, see Figure 4.3. In view of the way
we have derivedex– by considering the growth of debt, it should now be no surprise to
you that the function
y=exis often described as representing thelaw of natural growth. For example, it might
describe unrestrained bacterial growth. The function
y=e−x=(e−^1 )xon the other hand defines thelaw of natural decay. For example it describes the decrease
in mass of a radioactive element over time. Its graph is also shown in Figure 4.3.