We now pass on to a generalization of the idea of a series, which mathematics, true to its method, makes by use of the variable. Hitherto, we have only contemplated series in which each definite term was a definite number. But equally well we can generalize, and make each term to be some mathematical expression containing a variable $x$ . Thus we may consider the series $1$ , $x$ , $x^{2}$ , $x^{3}$ , , $x^{n}$ , , and the series $x, \frac{x^{2}}{2}, \frac{x^{3}}{3}, \dots, \frac{x^{n}}{n}, \dots .$

In order to symbolize the general idea of any such function, conceive of a function of $x$ , $f_{n} (x)$ say, which involves in its formation a variable integer $n$ , then, by giving $n$ the

values $1$ , $2$ , $3$ , etc., in succession, we get the series $f_{1} (x), f_{2} (x), f_{3} (x), \dots, f_{n} (x), \dots .$ Such a series may be convergent for some values of $x$ and divergent for others. It is, in fact, rather rare to find a series involving a variable $x$ which is convergent for all values of $x$ ,–-at least in any particular instance it is very unsafe to assume that this is the case. For example, let us examine the simplest of all instances, namely, the "geometrical"

series $1, x, x^{2}, x^{3}, \dots, x^{n}, \dots .$ The sum of $n$ terms is given by $s_{n} = 1 + x + x^{2} + x^{3} + \dots + x^{n} .$

Now multiply both sides by $x$ and we get $x s_{n} = x + x^{2} + x^{3} + x^{4} + \dots + x^{n} + x^{n + 1} .$ Now subtract the last line from the upper line and we get $s_{n} (1 - x) = s_{n} - x s_{n} = 1 - x^{n + 1},$ and hence (if $x$ be not equal to $1$ ) $s_{n} = \frac{1 - x^{n + 1}}{1 - x} = \frac{1}{1 - x} - \frac{x^{n + 1}}{1 - x} .$ Now if $x$ be numerically less than $1$ , for sufficiently large values of $n$ , $\frac{x^{n + 1}}{1 - x}$ is always numerically

less than $k$ , however $k$ be chosen. Thus, if $x$ be numerically less than $1$ , the series $1$ , $x$ , $x^{2}$ , , $x^{n}$ , is convergent, and $\frac{1}{1 - x}$ is its limit. This statement is symbolized by $\frac{1}{1 - x} = 1 + x + x^{2} + \dots + x^{n} + \dots, (- 1 < x < 1) .$ But if $x$ is numerically greater than $1$ , or numerically equal to $1$ , the series is divergent. In other words, if $x$ lie between $- 1$ and $+ 1$ , the series is convergent; but if $x$ be equal to $- 1$ or $+ 1$ , or if $x$ lie outside the interval $- 1$ to $+ 1$ , then the series is divergent. Thus the series is convergent at all "points" within the interval $- 1$ to $+ 1$ , exclusive of the end points.

At this stage of our enquiry another question arises. Suppose that the series $f_{1} (x) + f_{2} (x) + f_{3} (x) + \dots + f_{n} (x) + \dots$ is convergent for all values of $x$ lying within the interval $a$ to $b$ , i.e. the series is convergent for any value of $x$ which is greater than $a$ and less than $b$ . Also, suppose we want to be sure that in approximating to the limit we add together enough terms to come within some standard of approximation $k$ . Can we always state some number of terms, say $n$ , such that, if we take $n$ or more terms to form the sum, then whatever value $x$ has

within the interval we have satisfied the desired standard of approximation?

Sometimes we can and sometimes we cannot

do this for each value of $k$ . When we can, the series is called uniformly convergent throughout the interval, and when we cannot do so, the series is called non-uniformly convergent throughout the interval. It makes a great difference to the properties of a series whether it is or is not uniformly convergent through an interval. Let us illustrate the matter by the simplest example and the simplest numbers.