69. Functions of \(n\) which increase steadily with \(n\).
A special but particularly important class of functions of \(n\) is formed by those whose variation as \(n\) tends to \(\infty\) is always in the same direction, that is to say those which always increase (or always decrease) as \(n\) increases. Since \(-\phi(n)\) always increases if \(\phi(n)\) always decreases, it is not necessary to consider the two kinds of functions separately; for theorems proved for one kind can at once be extended to the other.
It is to be observed that we do not exclude the case in which \(\phi(n)\) has the same value for several values of \(n\); all we exclude is possible decrease. Thus the function \[\phi(n) = 2n + (-1)^{n},\] whose values for \(n = 0\), \(1\), \(2\), \(3\), \(4\), … are \[1,\ 1,\ 5,\ 5,\ 9,\ 9,\ \dots\] is said to increase steadily with \(n\). Our definition would indeed include even functions which remain constant from some value of \(n\) onwards; thus \(\phi(n) = 1\) steadily increases according to our definition. However, as these functions are extremely special ones, and as there can be no doubt as to their behaviour as \(n\) tends to \(\infty\), this apparent incongruity in the definition is not a serious defect.
There is one exceedingly important theorem concerning functions of this class.
Theorem. If \(\phi(n)\) steadily increases with \(n\), then either \(\phi(n)\) tends to a limit as \(n\) tends to \(\infty\), or \(\phi(n)\to +\infty\).
That is to say, while there are in general five alternatives as to the behaviour of a function, there are two only for this special kind of function.
This theorem is a simple corollary of Dedekind’s Theorem (§ 17). We divide the real numbers \(\xi\) into two classes \(L\) and \(R\), putting \(\xi\) in \(L\) or \(R\) according as \(\phi(n) \geq \xi\) for some value of \(n\) (and so of course for all greater values), or \(\phi(n) < \xi\) for all values of \(n\).
The class \(L\) certainly exists; the class \(R\) may or may not. If it does not, then, given any number \(\Delta\), however large, \(\phi(n) > \Delta\) for all sufficiently large values of \(n\), and so \[\phi(n) \to +\infty.\]
If on the other hand \(R\) exists, the classes \(L\) and \(R\) form a section of the real numbers in the sense of § 17. Let \(a\) be the number corresponding to the section, and let \(\epsilon\) be any positive number. Then \(\phi(n) < a + \epsilon\) for all values of \(n\), and so, since \(\epsilon\) is arbitrary, \(\phi(n) \leq a\). On the other hand \(\phi(n) > a – \epsilon\) for some value of \(n\), and so for all sufficiently large values. Thus \[a – \epsilon < \phi(n) \leq a\] for all sufficiently large values of \(n\); \[\phi(n)\to a.\]
It should be observed that in general \(\phi(n) < a\) for all values of \(n\); for if \(\phi(n)\) is equal to \(a\) for any value of \(n\) it must be equal to \(a\) for all greater values of \(n\). Thus \(\phi(n)\) can never be equal to \(a\) except in the case in which the values of \(\phi(n)\) are ultimately all the same. If this is so, \(a\) is the largest member of \(L\); otherwise \(L\) has no largest member.
COR 1. If \(\phi(n)\) increases steadily with \(n\), then it will tend to a limit or to \(+\infty\) according as it is or is not possible to find a number \(K\) such that \(\phi(n) < K\) for all values of \(n\).
We shall find this corollary exceedingly useful later on.
COR 2. If \(\phi(n)\) increases steadily with \(n\), and \(\phi(n) < K\) for all values of \(n\), then \(\phi(n)\) tends to a limit and this limit is less than or equal to \(K\).
It should be noticed that the limit may be equal to \(K\): if \(\phi(n) = 3 – (1/n)\), then every value of \(\phi(n)\) is less than \(3\), but the limit is equal to \(3\).
COR 3. If \(\phi(n)\) increases steadily with \(n\), and tends to a limit, then \[\phi(n) \leq \lim\phi(n)\] for all values of \(n\).
The reader should write out for himself the corresponding theorems and corollaries for the case in which \(\phi(n)\) decreases as \(n\) increases.
70.
The great importance of these theorems lies in the fact that they give us (what we have so far been without) a means of deciding, in a great many cases, whether a given function of \(n\) does or does not tend to a limit as \(n \to \infty\), without requiring us to be able to guess or otherwise infer beforehand what the limit is. If we know what the limit, if there is one, must be, we can use the test \[|\phi(n) – l| < \epsilon\quad (n \geq n_{0}):\] as for example in the case of \(\phi(n) = 1/n\), where it is obvious that the limit can only be zero. But suppose we have to determine whether \[\phi(n) = \left(1 + \frac{1}{n}\right)^{n}\] tends to a limit. In this case it is not obvious what the limit, if there is one, will be: and it is evident that the test above, which involves \(l\), cannot be used, at any rate directly, to decide whether \(l\) exists or not.
Of course the test can sometimes be used indirectly, to prove by means of a reductio ad absurdum that \(l\) cannot exist. If .eg. \(\phi(n) = (-1)^{n}\), it is clear that \(l\) would have to be equal to \(1\) and also equal to \(-1\), which is obviously impossible.
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