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

SERIES 213
These definitions are equivalent to the geometrical definitions, and both series can be
proved to be convergent for all values of x,
and uniformly convergent throughout any
interval. These series for sine and cosine
have a general likeness to the exponential
series given above. They are, indeed, intimately connected with it by means of the
theory of imaginary numbers explained in
Chapters VII and VIII.
Y/
A' o
Xi 0 x
Fig. 29.
The graph of the exponential function is
given in fig. 29. It cuts the axis OY at the
point y= 1, as evidently it ought to do, since
when x=O every term of the series except
the first is zero. The importance of the exponential function is that it represents any
changing physical quantity whose rate of
increase at any instant is a uniform percentage of its value at that instant. For