Python 3, 81 bytes
from fractions import* v=i=Fraction(1) while 1:v+=(v+1/i<=1!=print(i)or-1)/i;i+=1 -8 bytes thanks to ovs
hyperneutrino ♦
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from fractions import* v=i=Fraction(1) while 1:v+=(v+1/i<=1!=print(i)or-1)/i;i+=1 -8 bytes thanks to ovs