XIV

In the case of the geometric series $1+x+x^2+\dots +x^n+\dots$, a simple algebraic expression ${\displaystyle \frac{1}{1-x}}$ can be given for its limit in its interval of convergence. But this is not always the case. Often we can prove a series to be convergent within a certain interval, though we know nothing more about its limit except that it is the limit of the series.

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 $1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\dots +\frac{x^n}{n!}+\dots ,$ where $n!$ has the meaning defined earlier in this chapter. This series can be proved to be absolutely 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 $\exp x$. Thus, by definition, $\exp x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\dots +\frac{x^n}{n!}+\dots .$ $\exp x$ is called the exponential function.

It is fairly easy to prove, with a little knowledge of elementary mathematics, that $(\exp x)\times (\exp y)=\exp (x+y).\text{quadensuremath{(A)}\}$ In other words that

$$\begin{gathered} (\exp x)\times (\exp y) \\ =1+(x+y)+\frac{(x+y{)}^2}{2!}+\frac{(x+y{)}^3}{3!}+\dots +\frac{(x+y{)}^n}{n!}+\dots . \end{gathered}$$

This property (A) is an example of what is called an addition-theorem. When any

function [say $f(x)$] has been defined, the first thing we do is to try to express $f(x+y)$ in terms of known functions of $x$ only, and known functions of $y$ only. If we can do so, the result is called an addition-theorem. Addition-theorems play a great part in mathematical analysis. Thus the addition-theorem for the sine is given by $\sin (x+y)=\sin x\cos y+\cos x\sin y,$ and for the cosine by $\cos (x+y)=\cos x\cos y-\sin x\sin y.$

As a matter of fact the best ways of defining $\sin x$ and $\cos x$ are not by the elaborate geometrical methods of the previous chapter, but as the limits respectively of the series $x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\text{etc.}\dots ,$ and $1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\text{etc.}\dots ,$ so that we put

$$\begin{gathered} \sin x\&=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\text{etc.}\dots , \\ \cos x\&=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\text{etc.}\dots . \end{gathered}$$

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 [chapter:VII]VII. and [chapter:VIII]VIII. 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 $x=0$ 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