§2. Absolutely convergent series
107
since !(x) and x - 1 always have the same sign.
It remains to verify that the function ! is strictly increasing, i.e. that
o < x < x' implies !(x) < !(x'). But x' = xy with y > 1 and so !(x') =
!(x) + !(y). It is therefore enough to show that
(10.11)
y > 1 ¢:=* !(y) > 0,
which follows from (10), qed.
These results, which will be revisited in detail and by other methods in
Chap. IV, are to be compared with the remarks following Example 7 of nO 5.
We can now define the Napierian logarithm by the formula
(10.12)
logx = limn(x 1 / n -1) = lim Un.
The formulae (7) are then written
(10.13)
log(xy) = log x + logy,
log' x = l/x
where log' denotes the derivative of the function log. The relation (10) applied
to y / x shows that
1
log y - log x
1
- <
Y
y-x
x
As we shall see later, the functions log and exp are inverses of each other.
In other words,
(10.14)
exp(log x) = x,
log (expy) = y
for all x > 0 and y E R We can now show that these relations are plausible.
For n large, y = log x is "almost" equal to n(xl/n -1) and exp(y) "almost"
equal to (1 + y/n)n, by (6), "so" exp(logx) is "almost" equal to
[1 + n(xl/n -l)/nr = x,
"qed". Similarly, x = exp(y) is "almost" equal to (1 +y/n)n by (6), "so" xl/n
is "almost" equal to 1 + yin, "so" n(xl/n - 1) "is almost" equal to y, again
"qed" in quotes.
These arguments appear obvious only so long as one has not understood
the problems that they pose: there is no general theorem in analysis that will
allow us to legitimise them in the form, due to Halley the astronomer, and
to Euler, in which we have just presented them. A correct argument would
be, for example, to write
log (expx)
lim n {exp(x)l/n - 1} =
n ..... oo
lim n{[ lim (1 + x/m)m] l/n -I}
n--+oo
m-+cx>