Given the setting:
In a world where everyone wants a girl child, each family continues having babies till they have a girl. What do you think will the boy to girl ratio be eventually? (Assuming probability of having a boy or a girl is the same).
Is the following reasoning correct: the ratio of females in each family should be $\text{Exp}(\frac{1}{X}) =1-\ln(2)$ where $X\sim \text{Geo}(\frac{1}{2})$ since there is only one girl per family and variable amount of boys.
How can it be reconciled with the correct answer of a total uniform gender population ratio?
The expectation of $\frac1X$ (with $X\ge1$) is not suitable for this problem. $\frac1X$ represents the number of total children born per boy and takes fractional values, whereas what you really want is the number of boys born per girl, for which taking $E(X-1)=1$ suffices.
If you tried using $\frac1{X-1}$ to account for the final girl, you would divide by zero upon setting $X=1$.