What is the sum of the reciprocals of numbers with only $0$'s and $1$'s in their base $10$ expansion? $$x=\frac{1}{1}+\frac{1}{10}+\frac{1}{11}+\frac{1}{100}+\frac{1}{101}+\frac{1}{110}+\frac{1}{111}+\cdots.$$ Is the result transcendental, and does it have a closed form? I have figured out that $1.238405615301<x<1.238405615306$, but do not know where to go from there.
2026-03-28 19:39:56.1774726796
On a base 10 Infinite Series
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