Chapter IV
Limits of Functions of a Positive Integral Variable

50. Functions of a positive integral variable. In Chapter II we discussed the notion of a function of a real variable $x$, and illustrated the discussion by a large number of examples of such functions. And the reader will remember that there was one important particular with regard to which the functions which we took as illustrations differed very widely. Some were defined for all values of $x$, some for rational values only, some for integral values only, and so on.

Consider, for example, the following functions: (i) $x$, (ii) $\sqrt{x}$, (iii) the denominator of $x$, (iv) the square root of the product of the numerator and the denominator of $x$, (v) the largest prime factor of $x$, (vi) the product of $\sqrt{x}$ and the largest prime factor of $x$, (vii) the $x$th prime number, (viii) the height measured in inches of convict $x$ in Dartmoor prison.

Then the aggregates of values of $x$ for which these functions are defined or, as we may say, the fields of definition of the functions, consist of (i) all values of $x$, (ii) all positive values of $x$, (iii) all rational values of $x$, (iv) all positive rational values of $x$, (v) all integral values of $x$, (vi), (vii) all positive integral values of $x$, (viii) a certain number of positive integral values of $x$, viz., $1$, $2$, …, $N$, where $N$ is the total number of convicts at Dartmoor at a given moment of time.³⁶

Now let us consider a function, such as (vii) above, which is defined for all positive integral values of $x$ and no others. This function may be regarded from two slightly different points of view. We may consider it, as has so far been our custom, as a function of the real variable $x$ defined for some only of the values of $x$, viz. positive integral values, and say that for all other values of $x$ the definition fails. Or we may leave values of $x$ other than positive integral values entirely out of account, and regard our function as a function of the positive integral variable $n$, whose values are the positive integers

It is obvious that any function of $x$ defined for all values of $x$ gives rise to a function of $n$ defined for all values of $n$. Thus from the function $y=x^2$ we deduce the function $y=n^2$ by merely omitting from consideration all values of $x$ other than positive integers, and the corresponding values of $y$. On the other hand from any function of $n$ we can deduce any number of functions of $x$ by merely assigning values to $y$, corresponding to values of $x$ other than positive integral values, in any way we please.

51. Interpolation. The problem of determining a function of $x$ which shall assume, for all positive integral values of $x$, values agreeing with those of a given function of $n$, is of extreme importance in higher mathematics. It is called the problem of functional interpolation.

Were the problem however merely that of finding some function of $x$ to fulfil the condition stated, it would of course present no difficulty whatever. We could, as explained above, simply fill in the missing values as we pleased: we might indeed simply regard the given values of the function of $n$ as all the values of the function of $x$ and say that the definition of the latter function failed for all other values of $x$. But such purely theoretical solutions are obviously not what is usually wanted. What is usually wanted is some formula involving $x$ (of as simple a kind as possible) which assumes the given values for $x=1$, $2$, ….

In some cases, especially when the function of $n$ is itself defined by a formula, there is an obvious solution. If for example , where is a function of $n$, such as $n^2$ or $cosn\pi$, which would have a meaning even were $n$ not a positive integer, we naturally take our function of $x$ to be . But even in this very simple case it is easy to write down other almost equally obvious solutions of the problem. For example

In other cases may be defined by a formula, such as , which ceases to define for some values of $x$ (as here in the case of fractional values of $x$ with even denominators, or irrational values). But it may be possible to transform the formula in such a way that it does define for all values of $x$. In this case, for example,

In other cases may be defined for some values of $x$ other than positive integers, but not for all. Thus from $y=n^n$ we are led to $y=x^x$. This expression has a meaning for some only of the remaining values of $x$. If for simplicity we confine ourselves to positive values of $x$, then $x^x$ has a meaning for all rational values of $x$, in virtue of the definitions of fractional powers adopted in elementary algebra. But when $x$ is irrational $x^x$ has (so far as we are in a position to say at the present moment) no meaning at all. Thus in this case the problem of interpolation at once leads us to consider the question of extending our definitions in such a way that $x^x$ shall have a meaning even when $x$ is irrational. We shall see later on how the desired extension may be effected.

Again, consider the case in which

52. Finite and infinite classes. Before we proceed further it is necessary to make a few remarks about certain ideas of an abstract and logical nature which are of constant occurrence in Pure Mathematics.

In the first place, the reader is probably familiar with the notion of a class. It is unnecessary to discuss here any logical difficulties which may be involved in the notion of a ‘class’: roughly speaking we may say that a class is the aggregate or collection of all the entities or objects which possess a certain property, simple or complex. Thus we have the class of British subjects, or members of Parliament, or positive integers, or real numbers.

Moreover, the reader has probably an idea of what is meant by a finite or infinite class. Thus the class of British subjects is a finite class: the aggregate of all British subjects, past, present, and future, has a finite number $n$, though of course we cannot tell at present the actual value of $n$. The class of present British subjects, on the other hand, has a number $n$ which could be ascertained by counting, were the methods of the census effective enough.

On the other hand the class of positive integers is not finite but infinite. This may be expressed more precisely as follows. If $n$ is any positive integer, such as $1000$, $1,000,000$ or any number we like to think of, then there are more than $n$ positive integers. Thus, if the number we think of is $1,000,000$, there are obviously at least $1,000,001$ positive integers. Similarly the class of rational numbers, or of real numbers, is infinite. It is convenient to express this by saying that there are an infinite number of positive integers, or rational numbers, or real numbers. But the reader must be careful always to remember that by saying this we mean simply that the class in question has not a finite number of members such as $1000$ or $1,000,000$.

53. Properties possessed by a function of $n$ for large values of $n$. We may now return to the ‘functions of $n$’ which we were discussing in §§ 50–51. They have many points of difference from the functions of $x$ which we discussed in Chap. II. But there is one fundamental characteristic which the two classes of functions have in common: the values of the variable for which they are defined form an infinite class. It is this fact which forms the basis of all the considerations which follow and which, as we shall see in the next chapter, apply, mutatis mutandis, to functions of $x$ as well.

Suppose that is any function of $n$, and that $P$ is any property which may or may not have, such as that of being a positive integer or of being greater than $1$. Consider, for each of the values $n=1$, $2$, $3$, …, whether has the property $P$ or not. Then there are three possibilities:—

(a) may have the property $P$ for all values of $n$, or for all values of $n$ except a finite number $N$ of such values:

(b) may have the property for no values of $n$, or only for a finite number $N$ of such values:

(c) neither (a) nor (b) may be true.

If (b) is true, the values of $n$ for which has the property form a finite class. If (a) is true, the values of $n$ for which has not the property form a finite class. In the third case neither class is finite. Let us consider some particular cases.

(1) Let , and let $P$ be the property of being a positive integer. Then has the property $P$ for all values of $n$.

If on the other hand $P$ denotes the property of being a positive integer greater than or equal to $1000$, then has the property for all values of $n$ except a finite number of values of $n$, viz. $1$, $2$, $3$, …, $999$. In either of these cases (a) is true.

(2) If , and $P$ is the property of being less than $1000$, then (b) is true.

(3) If , and $P$ is the property of being odd, then (c) is true. For  is odd if $n$ is odd and even if $n$ is even, and both the odd and the even values of $n$ form an infinite class.

Example. Consider, in each of the following cases, whether (a), (b), or (c) is true:

(i) , $P$ being the property of being a perfect square,

(ii) , where $p_n$ denotes the $n$th prime number, $P$ being the
property of being odd,

(iii) , $P$ being the property of being even,

(iv) , $P$ being the property ,

(v) , $P$ being the property ,

(vi) , $P$ being the property ,

(vii) , $P$ being the property ,

(viii) , $P$ being the property ,

(ix) , $P$ being the property ,

(x) , or , $P$ being either of the properties or ,

(xi) , $P$ being the property .

54. Let us now suppose that  and $P$ are such that the assertion (a) is true, i.e. that  has the property $P$, if not for all values of $n$, at any rate for all values of $n$ except a finite number $N$ of such values. We may denote these exceptional values by

We shall frequently express this by saying that has the property for large, or very large, or all sufficiently large values of $n$. Thus when we say that  has the property $P$ (which will as a rule be a property expressed by some relation of inequality) for large values of $n$, what we mean is that we can determine some definite number, $n_0$ say, such that has the property for all values of $n$ greater than or equal to $n_0$. This number $n_0$, in the examples considered above, may be taken to be any number greater than $n_N$, the greatest of the exceptional numbers: it is most natural to take it to be $n_N+1$.

Thus we may say that ‘all large primes are odd’, or that ‘$1/n$ is less than $.001$ for large values of $n$’. And the reader must make himself familiar with the use of the word large in statements of this kind. Large is in fact a word which, standing by itself, has no more absolute meaning in mathematics than in the language of common life. It is a truism that in common life a number which is large in one connection is small in another; $6$ goals is a large score in a football match, but $6$ runs is not a large score in a cricket match; and $400$ runs is a large score, but £$400$ is not a large income: and so of course in mathematics large generally means large enough, and what is large enough for one purpose may not be large enough for another.

We know now what is meant by the assertion ‘ has the property $P$ for large values of $n$’. It is with assertions of this kind that we shall be concerned throughout this chapter.

55. The phrase ‘$n$ tends to infinity’. There is a somewhat different way of looking at the matter which it is natural to adopt. Suppose that $n$ assumes successively the values $1$, $2$, $3$, …. The word ‘successively’ naturally suggests succession in time, and we may suppose $n$, if we like, to assume these values at successive moments of time (e.g. at the beginnings of successive seconds). Then as the seconds pass $n$ gets larger and larger and there is no limit to the extent of its increase. However large a number we may think of (e.g. $2,147,483,647$), a time will come when $n$ has become larger than this number.

It is convenient to have a short phrase to express this unending growth of $n$, and we shall say that $n$ tends to infinity, or $n\to \infty$, this last symbol being usually employed as an abbreviation for ‘infinity’. The phrase ‘tends to’ like the word ‘successively’ naturally suggests the idea of change in time, and it is convenient to think of the variation of $n$ as accomplished in time in the manner described above. This however is a mere matter of convenience. The variable $n$ is a purely logical entity which has in itself nothing to do with time.

The reader cannot too strongly impress upon himself that when we say that $n$ ‘tends to $\infty$’ we mean simply that $n$ is supposed to assume a series of values which increase continually and without limit. There is no number ‘infinity’: such an equation as

(1) that $\infty$ by itself means nothing, although phrases containing it sometimes mean something,

(2) that in every case in which a phrase containing the symbol $\infty$ means something it will do so simply because we have previously attached a meaning to this particular phrase by means of a special definition.

Now it is clear that if  has the property $P$ for large values of $n$, and if $n$ ‘tends to $\infty$’, in the sense which we have just explained, then $n$ will ultimately assume values large enough to ensure that has the property $P$. And so another way of putting the question ‘what properties has for sufficiently large values of $n$?’ is ‘how does behave as $n$ tends to $\infty$?’

56. The behaviour of a function of $n$ as $n$ tends to infinity. We shall now proceed, in the light of the remarks made in the preceding sections, to consider the meaning of some kinds of statements which are perpetually occurring in higher mathematics. Let us consider, for example, the two following statements: (a) $1/n$ is small for large values of $n$, (b)  is nearly equal to $1$ for large values of $n$. Obvious as they may seem, there is a good deal in them which will repay the reader’s attention. Let us take (a) first, as being slightly the simpler.

We have already considered the statement ‘$1/n$ is less than $.01$ for large values of $n$’. This, we saw, means that the inequality $1/n<.01$ is true for all values of $n$ greater than some definite value, in fact greater than $100$. Similarly it is true that ‘$1/n$ is less than $.0001$ for large values of $n$’: in fact $1/n<.0001$ if $n>10,000$. And instead of $.01$ or $.0001$ we might take $.000001$ or $.00000001$, or indeed any positive number we like.

It is obviously convenient to have some way of expressing the fact that any such statement as ‘$1/n$ is less than $.01$ for large values of $n$’ is true, when we substitute for $.01$ any smaller number, such as $.0001$ or $.000001$ or any other number we care to choose. And clearly we can do this by saying that ‘however small $\epsilon$ may be (provided of course it is positive), then $1/n<\epsilon$ for sufficiently large values of $n$’. That this is true is obvious. For $1/n<\epsilon$ if $n>1/\epsilon$, so that our ‘sufficiently large’ values of $n$ need only all be greater than $1/\epsilon$. The assertion is however a complex one, in that it really stands for the whole class of assertions which we obtain by giving to $\epsilon$ special values such as $.01$. And of course the smaller $\epsilon$ is, and the larger $1/\epsilon$, the larger must be the least of the ‘sufficiently large’ values of $n$: values which are sufficiently large when $\epsilon$ has one value are inadequate when it has a smaller.

The last statement italicised is what is really meant by the statement (a), that $1/n$ is small when $n$ is large. Similarly (b) really means “if , then the statement ‘ for sufficiently large values of $n$’ is true whatever positive value $suchas.01or.0001$ we attribute to $\epsilon$”. That the statement (b) is true is obvious from the fact that .

There is another way in which it is common to state the facts expressed by the assertions (a) and (b). This is suggested at once by § 55. Instead of saying ‘$1/n$ is small for large values of $n$’ we say ‘$1/n$ tends to $0$ as $n$ tends to $\infty$’. Similarly we say that ‘ tends to $1$ as $n$ tends to $\infty$’: and these statements are to be regarded as precisely equivalent to (a) and (b). Thus the statements

are equivalent to one another and to the more formal statement

or to the still more formal statement

The number $n_0$ which occurs in the last statement is of course a function of $\epsilon$. We shall sometimes emphasize this fact by writing $n_0$ in the form .

The reader should imagine himself confronted by an opponent who questions the truth of the statement. He would name a series of numbers growing smaller and smaller. He might begin with $.001$. The reader would reply that $1/n<.001$ as soon as $n>1000$. The opponent would be bound to admit this, but would try again with some smaller number, such as $.0000001$. The reader would reply that $1/n<.0000001$ as soon as $n>10,000,000$: and so on. In this simple case it is evident that the reader would always have the better of the argument.

We shall now introduce yet another way of expressing this property of the function $1/n$. We shall say that ‘the limit of $1/n$ as $n$ tends to $\infty$ is $0$’, a statement which we may express symbolically in the form

57. Now let us consider a different example: let . Then ‘$n^2$ is large when $n$ is large’. This statement is equivalent to the more formal statements

And it is natural in this case to say that ‘$n^2$ tends to $\infty$ as $n$ tends to $\infty$’, or ‘$n^2$ tends to $\infty$ with $n$’, and to write

Finally consider the function . In this case is large, but negative, when $n$ is large, and we naturally say that ‘$-n^2$ tends to $-\infty$ as $n$ tends to $\infty$’ and write

But we must once more repeat that in all these statements the symbols $\infty$, $+\infty$, $-\infty$ mean nothing whatever by themselves, and only acquire a meaning when they occur in certain special connections in virtue of the explanations which we have just given.

58. Definition of a limit. After the discussion which precedes the reader should be in a position to appreciate the general notion of a limit. Roughly we may say that  tends to a limit $l$ as $n$ tends to $\infty$if  is nearly equal to $l$ when $n$ is large. But although the meaning of this statement should be clear enough after the preceding explanations, it is not, as it stands, precise enough to serve as a strict mathematical definition. It is, in fact, equivalent to a whole class of statements of the type ‘for sufficiently large values of $n$,  differs from $l$ by less than $\epsilon$’. This statement has to be true for $\epsilon =.01$ or $.0001$ or any positive number; and for any such value of $\epsilon$ it has to be true for any value of $n$ after a certain definite value , though the smaller $\epsilon$ is the larger, as a rule, will be this value .

We accordingly frame the following formal definition:

DEFINITION I. The function is said to tend to the limit $l$ as $n$ tends to $\infty$, if, however small be the positive number $\epsilon$,  differs from $l$ by less than $\epsilon$ for sufficiently large values of $n$; that is to say if, however small be the positive number $\epsilon$, we can determine a number  corresponding to $\epsilon$, such that  differs from $l$ by less than $\epsilon$ for all values of $n$ greater than or equal to .

It is usual to denote the difference between  and $l$, taken positively, by . It is equal to or to , whichever is positive, and agrees with the definition of the modulus of , as given in Chap. III, though at present we are only considering real values, positive or negative.

With this notation the definition may be stated more shortly as follows: ‘if, given any positive number, $\epsilon$, however small, we can find  so that when , then we say that  tends to the limit $l$ as $n$ tends to $\infty$, and write

Sometimes we may omit the ‘$n\to \infty$’; and sometimes it is convenient, for brevity, to write .

The reader will find it instructive to work out, in a few simple cases, the explicit expression of $n_0$ as a function of $\epsilon$. Thus if then $l=0$, and the condition reduces to $1/n<\epsilon$ for $n\geqq n_0$, which is satisfied if .³⁷ There is one and only one case in which the same $n_0$ will do for all values of $\epsilon$. If, from a certain value $N$ of $n$ onwards,  is constant, say equal to $C$, then it is evident that for $n\geqq N$, so that the inequality is satisfied for $n\geqq N$ and all positive values of $\epsilon$. And if for $n\geqq N$ and all positive values of $\epsilon$, then it is evident that when $n\geqq N$, so that  is constant for all such values of $n$.

59. The definition of a limit may be illustrated geometrically as follows. The graph of  consists of a number of points corresponding to the values $n=1$, $2$, $3$, ….

Draw the line $y=l$, and the parallel lines $y=l-\epsilon$, $y=l+\epsilon$ at distance $\epsilon$ from it. Then

if, when once these lines have been drawn, no matter how close they may be together, we can always draw a line $x=n_0$, as in the figure, in such a way that the point of the graph on this line, and all points to the right of it, lie between them. We shall find this geometrical way of looking at our definition particularly useful when we come to deal with functions defined for all values of a real variable and not merely for positive integral values.

60. So much for functions of $n$ which tend to a limit as $n$ tends to $\infty$. We must now frame corresponding definitions for functions which, like the functions $n^2$ or $-n^2$, tend to positive or negative infinity. The reader should by now find no difficulty in appreciating the point of

DEFINITION II. The function  is said to tend to $+\infty$(positive infinity) with $n$, if, when any number $\Delta$, however large, is assigned, we can determine  so that when ; that is to say if, however large $\Delta$ may be, for sufficiently large values of $n$.

Another, less precise, form of statement is ‘if we can make  as large as we please by sufficiently increasing $n$’. This is open to the objection that it obscures a fundamental point, viz. that  must be greater than $\Delta$ for all values of $n$ such that , and not merely for some such values. But there is no harm in using this form of expression if we are clear what it means.

When tends to $+\infty$ we write

61. Some points concerning the definitions. The reader should be careful to observe the following points.

(1) We may obviously alter the values of  for any finite number of values of $n$, in any way we please, without in the least affecting the behaviour of  as $n$ tends to $\infty$. For example $1/n$ tends to $0$ as $n$ tends to $\infty$. We may deduce any number of new functions from $1/n$ by altering a finite number of its values. For instance we may consider the function  which is equal to $3$ for $n=1$, $2$, $7$, $11$, $101$, $107$, $109$, $237$ and equal to $1/n$ for all other values of $n$. For this function, just as for the original function $1/n$, . Similarly, for the function  which is equal to $3$ if $n=1$, $2$, $7$, $11$, $101$, $107$, $109$, $237$, and to $n^2$ otherwise, it is true that .

(2) On the other hand we cannot as a rule alter an infinite number of the values of  without affecting fundamentally its behaviour as $n$ tends to $\infty$. If for example we altered the function $1/n$ by changing its value to $1$ whenever $n$ is a multiple of $100$, it would no longer be true that . So long as a finite number of values only were affected we could always choose the number $n_0$ of the definition so as to be greater than the greatest of the values of $n$ for which was altered. In the examples above, for instance, we could always take $n_0>237$, and indeed we should be compelled to do so as soon as our imaginary opponent of § 56 had assigned a value of $\epsilon$ as small as $3$ (in the first example) or a value of $\Delta$ as great as $3$ (in the second). But now however large $n_0$ may be there will be greater values of $n$ for which  has been altered.

(3) In applying the test of Definition I it is of course absolutely essential that we should have not merely when $n=n_0$ but when $n\geqq n_0$, i.e. for $n_0$ and for all larger values of $n$. It is obvious, for example, that, if  is the function last considered, then given $\epsilon$ we can choose $n_0$ so that when $n=n_0$: we have only to choose a sufficiently large value of $n$ which is not a multiple of $100$. But, when $n_0$ is thus chosen, it is not true that when $n\geqq n_0$: all the multiples of $100$ which are greater than $n_0$ are exceptions to this statement.

(4) If is always greater than $l$, we can replace by . Thus the test whether $1/n$ tends to the limit $0$ as $n$ tends to $\infty$ is simply whether $1/n<\epsilon$ when $n\geqq n_0$. If however , then $l$ is again $0$, but is sometimes positive and sometimes negative. In such a case we must state the condition in the form , for example, in this particular case, in the form .

(5) The limit $l$ may itself be one of the actual values of . Thus if for all values of $n$, it is obvious that . Again, if we had, in (2) and (3) above, altered the value of the function, when $n$ is a multiple of $100$, to $0$ instead of to $1$, we should have obtained a function which is equal to $0$ when $n$ is a multiple of $100$ and to $1/n$ otherwise. The limit of this function as $n$ tends to $\infty$ is still obviously zero. This limit is itself the value of the function for an infinite number of values of $n$, viz. all multiples of $100$.

On the other hand the limit itself need not $andingeneralwillnot$ be the value of the function for any value of $n$. This is sufficiently obvious in the case of . The limit is zero; but the function is never equal to zero for any value of $n$.

The reader cannot impress these facts too strongly on his mind. A limit is not a value of the function: it is something quite distinct from these values, though it is defined by its relations to them and may possibly be equal to some of them. For the functions

(6) A function may be always numerically very large when $n$ is very large without tending either to $+\infty$ or to $-\infty$. A sufficient illustration of this is given by . A function can only tend to $+\infty$ or to $-\infty$ if, after a certain value of $n$, it maintains a constant sign.

Examples XXIII. Consider the behaviour of the following functions of $n$ as $n$ tends to $\infty$:

1. , where $k$ is a positive or negative integer or rational fraction. If $k$ is positive, then $n^k$ tends to $+\infty$ with $n$. If $k$ is negative, then $lim{n}^k=0$. If $k=0$, then $n^k=1$ for all values of $n$. Hence $lim{n}^k=1$.

The reader will find it instructive, even in so simple a case as this, to write down a formal proof that the conditions of our definitions are satisfied. Take for instance the case of $k>0$. Let $\Delta$ be any assigned number, however large. We wish to choose $n_0$ so that $n^k>\Delta$ when $n\geqq n_0$. We have in fact only to take for $n_0$ any number greater than $\sqrt[k]{\Delta }$. If e.g. $k=4$, then $n^4>10,000$ when $n\geqq 11$, $n^4>100,000,000$ when $n\geqq 101$, and so on.

2. , where $p_n$ is the $n$th prime number. If there were only a finite number of primes then would be defined only for a finite number of values of $n$. There are however, as was first shown by Euclid, infinitely many primes. Euclid’s proof is as follows. If there are only a finite number of primes, let them be $1$, $2$, $3$, $5$, $7$, $11$, … $N$. Consider the number . This number is evidently not divisible by any of $2$, $3$, $5$, … $N$, since the remainder when it is divided by any of these numbers is $1$. It is therefore not divisible by any prime save $1$, and is therefore itself prime, which is contrary to our hypothesis.

It is moreover obvious that for all values of $n$ (save $n=1$, $2$, $3$). Hence .

3. Let  be the number of primes less than $n$. Here again .

4. , where $\alpha$ is any positive number. Here

5. If , then : and if , then . These conclusions are in no way affected by the fact that at first  is much larger than , being in fact larger until $n=1,000,000$.

6. , , . The first function tends to $0$, the second to $+\infty$, the third does not tend either to a limit or to $+\infty$.

7. , where $\theta$ is any real number. Here , since $\mid sinn\theta \pi \mid \leqq 1$, and .

8. , , where $a$ and $b$ are any real numbers.

9. . If $\theta$ is integral then for all values of $n$, and therefore .

Next let $\theta$ be rational, e.g. $\theta =p/q$, where $p$ and $q$ are positive integers. Let $n=aq+b$ where $a$ is the quotient and $b$ the remainder when $n$ is divided by $q$. Then . Suppose, for example, $p$ even; then, as $n$ increases from $0$ to $q-1$,  takes the values

The case in which $\theta$ is irrational is a little more difficult. It is discussed in the next set of examples.

62. Oscillating Functions.

DEFINITION. When does not tend to a limit, nor to $+\infty$, nor to $-\infty$, as $n$ tends to $\infty$, we say that oscillates as $n$ tends to $\infty$.

A function certainly oscillates if its values form, as in the case considered in the last example above, a continual repetition of a cycle of values. But of course it may oscillate without possessing this peculiarity. Oscillation is defined in a purely negative manner: a function oscillates when it does not do certain other things.

The simplest example of an oscillatory function is given by

DEFINITION. If oscillates as $n$ tends to $\infty$, then  will be said to oscillate finitely or infinitely according as it is or is not possible to assign a number $K$ such that all the values of  are numerically less than $K$, i.e.  for all values of $n$.

These definitions, as well as those of §§ 58 and 60, are further illustrated in the following examples.

Examples XXIV. Consider the behaviour as $n$ tends to $\infty$of the following functions:

1. , , , .

2. , .

3. $1,000,000-n$, .

4. . In this case the values of  are

5. . The second term oscillates infinitely, but the first is very much larger than the second when $n$ is large. In fact and is greater than any assigned value $\Delta$ if $n>1+\sqrt[]{\Delta +1}$. Thus . It should be observed that in this case  is always less than , so that the function progresses to infinity by a continual series of steps forwards and backwards. It does not however ‘oscillate’ according to our definition of the term.

6. , , .

7. $sinn\theta \pi$. We have already seen (Exs. XXIII. 9) that  oscillates finitely when $\theta$ is rational, unless $\theta$ is an integer, when , .

The case in which $\theta$ is irrational is a little more difficult. But it is not difficult to see that  still oscillates finitely. We can without loss of generality suppose $0<\theta <1$. In the first place . Hence  must oscillate finitely or tend to a limit. We shall consider whether the second alternative is really possible. Let us suppose that

Hence

The reader should consider with particular care the argument ‘$cosn\theta \pi cos\frac{1}{2}\theta \pi$ is very small, and this can only be the case if $cosn\theta \pi$ is very small’. Why, he may ask, should it not be the other factor $cos\frac{1}{2}\theta \pi$ which is ‘very small’? The answer is to be found, of course, in the meaning of the phrase ‘very small’ as used in this connection. When we say ‘ is very small’ for large values of $n$, we mean that we can choose $n_0$ so that  is numerically smaller than any assigned number, if $n\geqq n_0$. Such an assertion is palpably absurd when made of a fixed number such as $cos\frac{1}{2}\theta \pi$, which is not zero.

Prove similarly that $cosn\theta \pi$ oscillates finitely, unless $\theta$ is an even integer.

8. , $sinn\theta \pi +1$, $sinn\theta \pi +n$, .

9. $acosn\theta \pi +bsinn\theta \pi$, ${sin}^{2}n\theta \pi$, $a{cos}^{2}n\theta \pi +b{sin}^{2}n\theta \pi$.

10. .

11. $nsinn\theta \pi$. If $\theta$ is integral, then , . If $\theta$ is rational but not integral, or irrational, then  oscillates infinitely.

12. . In this case  tends to $+\infty$if $a$ and $b$ are both positive, but to $-\infty$ if both are negative. Consider the special cases in which $a=0$, $b>0$, or $a>0$, $b=0$, or $a=0$, $b=0$. If $a$ and $b$ have opposite signs  generally oscillates infinitely. Consider any exceptional cases.

13. . If $\theta$ is integral, then . Otherwise  oscillates finitely, as may be shown by arguments similar to though more complex than those used in Exs. XXIII. 9 and XXIV. 7.³⁸

14. . If $\theta$ has a rational value $p/q$, then $n!\theta$ is certainly integral for all values of $n$ greater than or equal to $q$. Hence . The case in which $\theta$ is irrational cannot be dealt with without the aid of considerations of a much more difficult character.

15. , , where $\theta$ is rational.

16. , .

17. , , .

18. The smallest prime factor of $n$. When $n$ is a prime, . When $n$ is even, . Thus oscillates infinitely.

19. The largest prime factor of $n$.

20. The number of days in the year $n$ A.D.

Examples XXV. 1. If and for all values of $n$, then .

2. If , and for all values of $n$, then .

3. If , then .

4. If tends to a limit or oscillates finitely, and when $n\geqq n_0$, then  tends to a limit or oscillates finitely.

5. If tends to $+\infty$, or to $-\infty$, or oscillates infinitely, and

6. ‘If  oscillates and, however great be $n_0$, we can find values of $n$ greater than $n_0$ for which , and values of $n$ greater than $n_0$ for which , then oscillates’. Is this true? If not give an example to the contrary.

7. If as $n\to \infty$, then also , $p$ being any fixed integer. [This follows at once from the definition. Similarly we see that if  tends to $+\infty$ or $-\infty$or oscillates so also does .]

8. The same conclusions hold (except in the case of oscillation) if $p$ varies with $n$ but is always numerically less than a fixed positive integer $N$; or if $p$ varies with $n$ in any way, so long as it is always positive.

9. Determine the least value of $n_0$ for which it is true that

10. Determine the least value of $n_0$ for which it is true that

11. Determine the least value of $n_0$ for which it is true that

[(a) : (b)  or , according as  is odd or even, i.e. .]

12. Determine the least value of $n_0$ such that

63. Some general theorems with regard to limits. A. The behaviour of the sum of two functions whose behaviour is known.

THEOREM I. If  and  tend to limits $a$, $b$, then tends to the limit $a+b$.

This is almost obvious.³⁹ The argument which the reader will at once form in his mind is roughly this: ‘when $n$ is large,  is nearly equal to $a$ and to $b$, and therefore their sum is nearly equal to $a+b$’. It is well to state the argument quite formally, however.

Let $\epsilon$ be any assigned positive number (e.g. $.001$, $.0000001$, …). We require to show that a number $n_0$ can be found such that

when $n\geqq n_0$. Now by a proposition proved in Chap. III (more generally indeed than we need here) the modulus of the sum of two numbers is less than or equal to the sum of their moduli. Thus

when $n\geqq n_0$. But this is certainly the case. For since we can, by the definition of a limit, find $n_1$ so that when $n\geqq n_1$, and this however small $\epsilon \prime$ may be. Nothing prevents our taking $\epsilon \prime =\frac{1}{2}\epsilon$, so that when $n\geqq n_1$. Similarly we can find $n_2$ so that when $n\geqq n_2$. Now take $n_0$ to be the greater of the two numbers $n_1$, $n_2$. Then and when $n\geqq n_0$, and therefore (2) is satisfied and the theorem is proved.

The argument may be concisely stated thus: since and , we can choose $n_1$, $n_2$ so that

64. Results subsidiary to Theorem I. The reader should have no difficulty in verifying the following subsidiary results.

1. If  tends to a limit, but  tends to $+\infty$ or to $-\infty$ or oscillates finitely or infinitely, then behaves like .

2. If , and or oscillates finitely, then .

In this statement we may obviously change $+\infty$ into $-\infty$ throughout.

3. If and , then may tend either to a limit or to $+\infty$ or to $-\infty$ or may oscillate either finitely or infinitely.

These five possibilities are illustrated in order by (i) , , (ii) , , (iii) , , (iv) , , (v) , . The reader should construct additional examples of each case.

4. If and  oscillates infinitely, then may tend to $+\infty$ or oscillate infinitely, but cannot tend to a limit, or to $-\infty$, or oscillate finitely.

For ; and, if behaved in any of the three last ways, it would follow, from the previous results, that , which is not the case. As examples of the two cases which are possible, consider (i) , , (ii) , . Here again the signs of $+\infty$ and $-\infty$ may be permuted throughout.

5. If and both oscillate finitely, then must tend to a limit or oscillate finitely.

As examples take

6. If oscillates finitely, and  infinitely, then oscillates infinitely.

For is in absolute value always less than a certain constant, say $K$. On the other hand , since it oscillates infinitely, must assume values numerically greater than any assignable number (e.g. $10K$, $100K$, …). Hence must assume values numerically greater than any assignable number (e.g. $9K$, $99K$, …). Hence must either tend to $+\infty$ or $-\infty$or oscillate infinitely. But if it tended to $+\infty$ then

7. If both and oscillate infinitely, then may tend to a limit, or to $+\infty$, or to $-\infty$, or oscillate either finitely or infinitely.

Suppose, for instance, that , while  is in turn each of the functions , , , , . We thus obtain examples of all five possibilities.

The results 1–7 cover all the cases which are really distinct. Before passing on to consider the product of two functions, we may point out that the result of Theorem I may be immediately extended to the sum of three or more functions which tend to limits as $n\to \infty$.

65. B. The behaviour of the product of two functions whose behaviour is known. We can now prove a similar set of theorems concerning the product of two functions. The principal result is the following.

THEOREM II. If and , then

Let

The following is a strictly formal proof. We have

We need hardly point out that this theorem, like Theorem I, may be immediately extended to the product of any number of functions of $n$. There is also a series of subsidiary theorems concerning products analogous to those stated in § 64 for sums. We must distinguish now six different ways in which  may behave as $n$ tends to $\infty$. It may (1) tend to a limit other than zero, (2) tend to zero, (3a) tend to $+\infty$, (3b) tend to $-\infty$, (4) oscillate finitely, (5) oscillate infinitely. It is not necessary, as a rule, to take account separately of (3a) and (3b), as the results for one case may be deduced from those for the other by a change of sign.