Are there any nice examples of infinite sequences of irrational numbers converging to rational numbers?
One idea I had was the sequence: $ 0.1001000010000001\cdots,0.1101000110000001\cdots,\cdots,0.1111000110000001\cdots,$ etc.
Where the first term in the sequence has ones in the place $i^2$ positions to the right of the decimal point. $(i=1,2,3,\dots)$ For the second term, we keep all the ones from the first term and add one in place of the first zero after the decimal point. We then add ones $i^3$ places after the decimal point. (The logical sum 1+1=1, i.e a number $i^2=j^3$ spaces after the decimal place has the value 1).
It is clear that this will converge to $1/9$, and I don't think the decimal expansion repeats at all.