But when we approximate by gradually

adding the successive terms of an infinite series, what are we approximating to? The difficulty is that the series has no "sum" in the straightforward sense of the word, because the operation of adding together its terms can never be completed. The answer is that we are approximating to the limit of the summation of the series, and we must now

proceed to explain what the "limit" of a series is.

The summation of a series approximates to a limit when the sum of any number of its terms, provided the number be large enough, is as nearly equal to the limit as you care to approach. But this description of the meaning of approximating to a limit evidently will not stand the vigorous scrutiny of modern mathematics. What is meant by large enough, and by nearly equal, and by care to approach? All these vague phrases must be explained in terms of the simple abstract ideas which alone are admitted into pure mathematics.

Let the successive terms of the series be $u_{1}$ , $u_{2}$ , $u_{3}$ , $u_{4}$ , , $u_{n}$ , etc., so that $u_{n}$ is the $n$ th term of the series. Also let $s_{n}$ be the sum of the $1$ st $n$ terms, whatever $n$ may be. So that:–-

$s_{1} = u_{1}, s_{2} = u_{1} + u_{2}, s_{3} = u_{1} + u_{2} + u_{3}, and s_{n} = u_{1} + u_{2} + u_{3} + \dots + u_{n} .$

Then the terms $s_{1}$ , $s_{2}$ , $s_{3}$ , , $s_{n}$ , form a new series, and the formation of this series is the process of summation of the original series. Then the "approximation" of the summation of the original series to a "limit" means the "approximation of the terms of this new series to a limit." And we have

now to explain what we mean by the approximation to a limit of the terms of a series.

Now, remembering the definition (given in [chapter]XII.) of a standard of approximation,

the idea of a limit means this: $l$ is the limit of the terms of the series $s_{1}$ , $s_{2}$ , $s_{3}$ , , $s_{n}$ , , if, corresponding to each real number $k$ , taken as a standard of approximation, a term $s_{n}$ of the series can be found so that all succeeding terms (i.e. $s_{n + 1}$ , $s_{n + 2}$ , etc.) approximate to $l$ within that standard of approximation. If another smaller standard $k^{1}$ be chosen, the term $s_{n}$ may be too early in the series, and a later term $s_{m}$ with the above property will then be found.

If this property holds, it is evident that as you go along to series $s_{1}$ , $s_{2}$ , $s_{3}$ , , $s_{n}$ , from left to right, after a time you come to terms all of which are nearer to $l$ than any number which you may like to assign. In other words you approximate to $l$ as closely as you like. The close connection of this definition of the limit of a series with the definition of a continuous function given in [chapter]XI. will be immediately perceived.

Then coming back to the original series $u_{1}$ , $u_{2}$ , $u_{3}$ , , $u_{n}$ , , the limit of the terms of the series $s_{1}$ , $s_{2}$ , $s_{3}$ , , $s_{n}$ , , is called the "sum to infinity" of the original series. But it is evident that this use of the word

"sum" is very artificial, and we must not assume the analogous properties to those of the ordinary sum of a finite number of terms without some special investigation.

Let us look at an example of a "sum to infinity." Consider the recurring decimal .1111 … . This decimal is merely a way of symbolizing the "sum to infinity" of the series .1 , .01 , .001 , .0001 , etc. The corresponding