View source for Liouville number

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A real number $\alpha$ such that for any $\nu\geq1$ the inequality

$$\left|\alpha-\frac pq\right|<q^{-\nu}$$

has infinitely many integer solutions $p$ and $q$ satisfying the conditions $q>0$, $(p,q)=1$. The fact that a Liouville number is transcendental (cf. [[Transcendental number|Transcendental number]]) follows from the Liouville theorem (cf. [[Liouville theorems|Liouville theorems]]). These numbers were studied by J. Liouville [[#References|[1]]].

Examples of Liouville numbers are:

$$\alpha_1=\sum_{n=1}^\infty2^{-n!},$$

$$\alpha_2=\sum_{n=1}^\infty(-1)^n2^{-3^n},$$

$$\alpha_3=\sum_{n=1}^\infty(10^{n!})^{-1}.$$

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Liouville,   "Sur des classes très étendues de quantités dont la valeur n'est ni algébrique, ni même réductible à des irrationelles algébriques"  ''C.R. Acad. Sci. Paris'' , '''18'''  (1844)  pp. 883–885</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.O. Gel'fond,   "Transcendental and algebraic numbers" , Dover, reprint  (1960)  (Translated from Russian)</TD></TR></table>



====Comments====


====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  O. Perron,   "Die Lehre von den Kettenbrüchen" , '''1''' , Teubner  (1977)  pp. Sect. 35</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  O. Perron,   "Irrationalzahlen" , Chelsea, reprint  (1948)</TD></TR>
</table>

[[Category:Number theory]]