I am just wondering - is there an iteration formula that converges to 1 recurring?

(not 0.11111111111 but 11111111111)

Not simply by saying 111111111 = infinity and then find one that keeps expanding, Im looking for

x0=1

x1=11

x2=111

etc.

Bit of a wierd question I know - But if anyone could show me a way (although I doubt that its possible) Id be very thankful Thanks