It will be convenient to prove at this stage a number of elementary inequalities which will be useful to us later on.

(i) It is evident that if \(\alpha > 1\) and \(r\) is a positive integer then \[r\alpha^{r} > \alpha^{r-1} + \alpha^{r-2} + \dots + 1.\] Multiplying both sides of this inequality by \(\alpha – 1\), we obtain \[r\alpha^{r}(\alpha – 1) > \alpha^{r} – 1;\] and adding \(r(\alpha^{r} – 1)\) to each side, and dividing by \(r(r + 1)\), we obtain \[\begin{equation*} \frac{\alpha^{r+1} – 1}{r + 1} > \frac{\alpha^{r} – 1}{r}\quad (\alpha > 1). \tag{1} \end{equation*}\] Similarly we can prove that \[\begin{equation*} \frac{1 – \beta^{r+1}}{r + 1} < \frac{1 – \beta^{r}}{r}\quad (0 < \beta < 1). \tag{2} \end{equation*}\]

It follows that if \(r\) and \(s\) are positive integers, and \(r > s\), then \[\begin{equation*} \frac{\alpha^{r} – 1}{r} > \frac{a^{s} – 1}{s},\quad \frac{1 – \beta^{r}}{r} < \frac{1 – \beta^{s}}{s}. \tag{3} \end{equation*}\] Here \(0 < \beta < 1 < \alpha\). In particular, when \(s = 1\), we have \[\begin{equation*} \alpha^{r} – 1 > r(\alpha – 1),\quad 1 – \beta^{r} < r(1 – \beta). \tag{4} \end{equation*}\]

(ii) The inequalities (3) and (4) have been proved on the supposition that \(r\) and \(s\) are positive integers. But it is easy to see that they hold under the more general hypothesis that \(r\) and \(s\) are any positive rational numbers. Let us consider, for example, the first of the inequalities (3). Let \(r = a/b\), \(s = c/d\), where \(a\), \(b\), \(c\), \(d\) are positive integers; so that \(ad > bc\). If we put \(\alpha = \gamma^{bd}\), the inequality takes the form \[(\gamma^{ad} – 1)/ad > (\gamma^{bc} – 1)/bc;\] and this we have proved already. The same argument applies to the remaining inequalities; and it can evidently be proved in a similar manner that \[\begin{equation*} \alpha^{s} – 1 < s(\alpha – 1),\quad 1 – \beta^{s} > s(1 – \beta), \tag{5} \end{equation*}\] if \(s\) is a positive rational number less than \(1\).

(iii) In what follows it is to be understood that all the letters denote positive numbers, that \(r\) and \(s\) are rational, and that \(\alpha\) and \(r\) are greater than \(1\), \(\beta\) and \(s\) less than \(1\). Writing \(1/\beta\) for \(\alpha\), and \(1/\alpha\) for \(\beta\), in (4), we obtain \[\begin{equation*} \alpha^{r} – 1 < r\alpha^{r-1}(\alpha – 1),\quad 1 – \beta^{r} > r\beta^{r-1}(1 – \beta). \tag{6} \end{equation*}\] Similarly, from (5), we deduce \[\begin{equation*} \alpha^{s} – 1 > s\alpha^{s-1}(\alpha – 1),\quad 1 – \beta^{s} < s\beta^{s-1}(1 – \beta). \tag{7} \end{equation*}\]

Combining (4) and (6), we see that \[\begin{equation*} r\alpha^{r-1}(\alpha – 1) > \alpha^{r} – 1 > r(\alpha – 1). \tag{8} \end{equation*}\] Writing \(x/y\) for \(\alpha\), we obtain \[\begin{equation*} rx^{r-1} (x – y) > x^{r} – y^{r} > ry^{r-1} (x – y) \tag{9} \end{equation*}\] if \(x > y > 0\). And the same argument, applied to (5) and (7), leads to \[\begin{equation*} sx^{s-1} (x – y) < x^{s} – y^{s} < sy^{s-1} (x – y). \tag{10} \end{equation*}\]

Example XXVIII

1. Verify (9) for \(r = 2\), \(3\), and (10) for \(s = \frac{1}{2}\), \(\frac{1}{3}\).

2. Show that (9) and (10) are also true if \(y > x > 0\).

3. Show that (9) also holds for \(r < 0\). [See Chrystal’s Algebra, vol. ii, pp. 43–45.]

4. If \(\phi(n) \to l\), where \(l > 0\), as \(n \to \infty\), then \(\phi^{k} \to l^{k}\), \(k\) being any rational number.

[We may suppose that \(k > 0\), in virtue of Theorem III of § 66; and that \(\frac{1}{2}l < \phi < 2l\), as is certainly the case from a certain value of \(n\) onwards. If \(k > 1\), \[k\phi^{k-1}(\phi – l) > \phi^{k} – l^{k} > kl^{k-1}(\phi – l)\] or \[kl^{k-1}(l – \phi) > l^{k} – \phi^{k} > k\phi^{k-1}(l – \phi),\] according as \(\phi > l\) or \(\phi < l\). It follows that the ratio of \(|\phi^{k} – l^{k}|\) and \(|\phi – l|\) lies between \(k(\frac{1}{2}l)^{k-1}\) and \(k(2l)^{k-1}\). The proof is similar when \(0 < k < 1\). The result is still true when \(l = 0\), if \(k > 0\).]

5. Extend the results of Exs. XXVII. 7, 8, 9 to the case in which \(r\) or \(k\) are any rational numbers.


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