The Seven and Seven Conjecture

When I was thirteen, sitting in the hallway between classes in what felt like an interminable school day, the following occurred to me.


I think it’s true, having tried it to as many decimal places as I could muster the energy for a couple decades ago. Not enough training in number theory to generate a real proof for it, though.  (Maybe more striaghtforward to prove that the series without the sevens converges on 1/49?)  Not enough historical knowledge to say if it’s just a spin on another idea already out there.  But there it is, I guess.