Fractions and Decimals

The rational number $< 1$ that will expand to the sequence $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ repeated infinitely many times when converted to decimal form is $\large{\frac{13717421}{1111111111}}$ $\rightarrow 13717421+1111111111=\boxed{1124828532}$ .

Proof:

Let $S = 0.\overline{0123456789}$ . Then $10^{10}*S = 123456789.\overline{0123456789}$ .

Hence by subtraction, $(10^{10}-1)S = 123456789 \rightarrow S = \large{\frac{123456789}{9999999999}=\frac{41152263}{3333333333}=\frac{13717421}{1111111111}}$ .