**Definition.** When \(\phi(n)\) does not tend to a limit, nor to \(+\infty\), nor to \(-\infty\), as \(n\) tends to \(\infty\), we say that \(\phi(n)\) **oscillates** as \(n\) tends to \(\infty\).

A function \(\phi(n)\) 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 \[\phi(n) = (-1)^{n},\] which is equal to \(+1\) when \(n\) is even and to \(-1\) when \(n\) is odd. In this case the values recur cyclically. But consider \[\phi(n) = (-1)^{n} + (1/n),\] the values of which are \[-1 + 1,\quad 1 + (1/2),\quad -1 + (1/3),\quad 1 + (1/4),\quad -1 + (1/5),\ \dots.\] When \(n\) is large every value is nearly equal to \(+1\) or \(-1\), and obviously \(\phi(n)\) does not tend to a limit or to \(+\infty\) or to \(-\infty\), and therefore it oscillates: but the values do not recur. It is to be observed that in this case every value of \(\phi(n)\) is numerically less than or equal to \(3/2\). Similarly \[\phi(n) = (-1)^{n} 100 + (1000/n)\] oscillates. When \(n\) is large, every value is nearly equal to \(100\) or to \(-100\). The numerically greatest value is \(900\) (for \(n = 1\)). But now consider \(\phi(n) = (-1)^{n}n\), the values of which are \(-1\), \(2\), \(-3\), \(4\), \(-5\), …. This function oscillates, for it does not tend to a limit, nor to \(+\infty\), nor to \(-\infty\). And in this case we cannot assign any limit beyond which the numerical value of the terms does not rise. The distinction between these two examples suggests a further definition.

**Definition.** If \(\phi(n)\) oscillates as \(n\) tends to \(\infty\), then \(\phi(n)\) 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 \(\phi(n)\) are numerically less than \(K\), \(|\phi(n)| < K\) for all values of \(n\).

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

- See Bromwich’s
*Infinite Series*, p. 485.↩︎

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