Calculus: Analytic Geometry and Calculus, with Vectors

504 Exponential and logarithmic functions

Since fi(al) = 1, this gives the remarkable parade` of inequalities

(8) 1 G

Cal

+ Qn\ 2

G

Cal

+ a2 + a3\1

(al

+ a2 + a3 +

\ 2 /J\ 3 /J 4 / G

a,a2^3 ala2a3 alaoa3aq

The denominator in these quotients cannot exceed the numerators, and therefore

(9)

,, ... a dal + a2 + +a.

(^12) ,, (n = 1,2,3, ..),
The simple corollary (9) of the more spectacular result (8) is a statement of a
very famous theorem which says that if a1, a2, , an, are positive numbers,
then their geometric mean is less than or equal to their arithmetic mean. Our work
enables us to show that equality holds only when the numbers a1, a2, ' ' , an are
equal. Many proofs of this theorem are known, and it is quite appropriate to
become interested in the matter by looking at the special case
(10) 27ala2as < (a, + a2 + a3)3
and seeking ways to prove it. Sometimes scientists say that boys work with
equalities and men work with inequalities.
23 For those who are interested in the matter, we present a direct proof that
lz
(1) lim f(x) = e when f(x) _ (1 + -/.
I=I X
When n is a positive integer, putting a = 1 and b = 1/n in the binomial formula
(2) (a + b)n = anb° + van-1bl + n(n2- 1) an-2b2
+n(n - 13 (n - 2)an-3P + ... +n(n - 1) (n

2) ... 2.1 a°b"

gives

(3) 11)+3!(i

)n)

• (1 /\1 )(1 )+

+ )

Hence

(4) f(n)51+1+2i+3 +...+ni

2 22 2n

It follows from Theorem 5.65 that the series in

(5) e

converges to a number which we can call e and that (5) holds. If m > n, then

replacing n by m in (3) shows that f(m) consists of (m + 1) terms of which the

first (n + 1) equal or exceed those in the right member of (3) and the remaining

m - n terms are positive. Hence f(m) > f(n) when m > n. As we can see