An introduction to mathematics, by A. N. Whitehead.

SERIES 211
But this is a very good way of defining a function; viz. as the limit of an infinite convergent series, and is, in fact, the way in which
most functions are, or ought to be, defined.
Thus, the most important series in elementary analysis is
X2 X3 Xn
1+x+-+-.+. + - +.
l+x+ + + 3! n!
where n! has the meaning defined earlier in
this chapter. This series can be proved to
be convergent for all values of x, and to be
uniformly convergent within any interval
which we like to take. Hence it has all the
comfortable mathematical properties which
a series should have. It is called the exponential series. Denote its sum to infinity
by expx. Thus, by definition,
X2 X3 Xn
expx=l+x+ -+-+... + +..
3! n!
expx is called the exponential function.
It is fairly easy to prove, with a little
knowledge of elementary mathematics, that
(expx) X (expy) =exp(x +y)... (A)
In other words that
(expx) X (expy) =
1 + (x +y) + (x +y)2 + (x +y)3 + +
l ^ ( y+ )... +
n! 3!
(X +y),
n!