It is called repeating decimal or recurring decimal, so you were very very close. The infinity sign is not used - instead there should be a horizontal line above the digits that are repeating ( not easily done in html ;-) ).
Rational numbers (M/N, where M and N are integral) do have this property when they are written in the usual decimal notation. Look these examples:
1.2 = 12/10
1.23 = 123/100
1.234 = 1234/1000
These "normal" decimal numbers can be expressed as fractions, but note that there always is a power of 10 in denominator. Our usual decimal notation can be considered just a shortcut of the full M/N notation.
Every fractional number (that cannot be further simplified) that has something else than a product of 2s and 5s (10=2*5) in the denominator will have infinite number of repeating digits when written in decimal notation.
1/3, 1/6, 1/7, 1/9, 1/11, 1/13, ...
Here is a provocative number: 1/2 = 0.5 = 0.500000000000000000000000000000000000...