202. Scales of infinity

The logarithmic scale. Let us consider once more the series of functions \[x,\quad \sqrt{x},\quad \sqrt[3]{x},\ \dots,\quad \sqrt[n]{x},\ \dots,\] which possesses the property that, if $f(x)$ and $\phi(x)$ are any two of the functions contained in it, then $f(x)$ and $\phi(x)$ both tend to $\infty$ as $x \to \infty$, while $f(x)/\phi(x)$ tends to $0$ or to $\infty$ according as $f(x)$ occurs to the right or the left of $\phi(x)$ in the series. We can now continue this series by the insertion of new terms to the right of all those already written down. We can begin with $\log x$, which tends to infinity more slowly than any of the old terms. Then $\sqrt{\log x}$ tends to $\infty$ more slowly than $\log x$, $\sqrt[3]{\log x}$ than $\sqrt{\log x}$, and so on. Thus we obtain a series \[x,\quad \sqrt{x},\quad \sqrt[3]{x},\ \dots,\quad \sqrt[n]{x},\ \dots\quad \log x,\quad \sqrt{\log x},\quad \sqrt[3]{\log x},\ \dots\quad \sqrt[n]{\log x},\ \dots\] formed of two simply infinite series arranged one after the other. But this is not all. Consider the function $\log\log x$, the logarithm of $\log x$. Since $(\log x)/x^{\alpha} \to 0$, for all positive values of $\alpha$, it follows on putting $x = \log y$ that \[(\log\log y)/(\log y)^{\alpha} = (\log x)/x^{\alpha} \to 0.\] Thus $\log\log y$ tends to $\infty$ with $y$, but more slowly than any power of $\log y$. Hence we may continue our series in the form \[\begin{gathered} x,\quad \sqrt{x},\quad \sqrt[3]{x},\ \dots\qquad \log x,\quad \sqrt{\log x},\quad \sqrt[3]{\log x},\ \dots\\ \log\log x,\quad \sqrt{\log\log x},\ \dots\quad \sqrt[n]{\log\log x},\ \dots;\end{gathered}\] and it will by now be obvious that by introducing the functions $\log\log\log x$, $\log\log\log\log x$, … we can prolong the series to any extent we like. By putting $x = 1/y$ we obtain a similar scale of infinity for functions of $y$ which tend to $\infty$ as $y$ tends to $0$ by positive values.¹

Example LXXXIV

1. Between any two terms $f(x)$, $F(x)$ of the series we can insert a new term $\phi(x)$ such that $\phi(x)$ tends to $\infty$ more slowly than $f(x)$ and more rapidly than $F(x)$. [Thus between $\sqrt{x}$ and $\sqrt[3]{x}$ we could insert $x^{5/12}$: between $\sqrt{\log x}$ and $\sqrt[3]{\log x}$ we could insert $(\log x)^{5/12}$. And, generally, $\phi(x) = \sqrt{f(x) F(x)}$ satisfies the conditions stated.]

2. Find a function which tends to $\infty$ more slowly than $\sqrt{x}$, but more rapidly than $x^{\alpha}$, where $\alpha$ is any rational number less than $1/2$. [$\sqrt{x}/(\log x)$ is such a function; or $\sqrt{x}/(\log x)^{\beta}$, where $\beta$ is any positive rational number.]

3. Find a function which tends to $\infty$ more slowly than $\sqrt{x}$, but more rapidly than $\sqrt{x}/(\log x)^{\alpha}$, where $\alpha$ is any rational number. [The function $\sqrt{x}/(\log\log x)$ is such a function. It will be gathered from these examples that incompleteness is an inherent characteristic of the logarithmic scale of infinity.]

4. How does the function \[f(x) = \{x^{\alpha} (\log x)^{\alpha’} (\log\log x)^{\alpha”}\}/ \{x^{\beta} (\log x)^{\beta’} (\log\log x)^{\beta”}\}\] behave as $x$ tends to $\infty$? [If $\alpha \neq \beta$ then the behaviour of \[f(x) = x^{\alpha-\beta} (\log x)^{\alpha’-\beta’} (\log\log x)^{\alpha”-\beta”}\] is dominated by that of $x^{\alpha-\beta}$. If $\alpha = \beta$ then the power of $x$ disappears and the behaviour of $f(x)$ is dominated by that of $(\log x)^{\alpha’-\beta’}$, unless $\alpha’ = \beta’$, when it is dominated by that of $(\log\log x)^{\alpha”-\beta”}$. Thus $f(x) \to \infty$ if $\alpha > \beta$, or $\alpha = \beta$, $\alpha’ > \beta’$, or $\alpha = \beta$, $\alpha’ = \beta’$, $\alpha” > \beta”$, and $f(x) \to 0$ if $\alpha < \beta$, or $\alpha = \beta$, $\alpha’ < \beta’$, or $\alpha = \beta$, $\alpha’ = \beta’$, $\alpha” < \beta”$.]

5. Arrange the functions $x/\sqrt{\log x}$, $x\sqrt{\log x}/\log\log x$, $x\log\log x/\sqrt{\log x}$, $(x\log\log\log x)/\sqrt{\log\log x}$ according to the rapidity with which they tend to infinity as $x \to \infty$.

6. Arrange \[\log\log x/(x\log x),\quad (\log x)/x,\quad x\log\log x/\sqrt{x^{2} + 1},\quad \{\sqrt{x + 1}\}/x(\log x)^{2}\] according to the rapidity with which they tend to zero as $x \to \infty$.

7. Arrange \[x\log\log(1/x),\quad \sqrt{x}/\{\log(1/x)\},\quad \sqrt{x\sin x\log(1/x)},\quad (1 – \cos x)\log(1/x)\] according to the rapidity with which they tend to zero as $x \to +0$.

8. Show that \[D_{x}\log\log x = 1/(x\log x),\quad D_{x}\log\log\log x = 1/(x\log x\log\log x),\] and so on.

9. Show that \[D_{x}(\log x)^{\alpha} = \alpha/\{x(\log x)^{1-\alpha}\},\quad D_{x}(\log\log x)^{\alpha} = \alpha/\{x\log x(\log\log x)^{1-\alpha}\},\] and so on.

For fuller information as to ‘scales of infinity’ see the author’s tract ‘Orders of Infinity’, Camb. Math. Tracts, No. 12.↩︎

$\leftarrow$ 199–201. The behaviour of $\log x$ as $x$ tends to infinity or to zero

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203. The number $e$ $\rightarrow$