The ratio of boys to girls will stay the same

When the chance of having a girl is G, the chance of having a boy is (1-G).

The chance of having only a boy is (1-G)
The chance of having a girl and a boy is(1-G)G
The chance of having 2 girls and a boy is (1-G)(G^2)
The chance of having N girls and a boy is (1-G)(G^N)

Now we have to multiply the probability with the number of boys/girls and add them together the get the average number of boys/girls.

For boys we get the following sum: (1-G) + (1-G)G + (1-G)(G^2) + …. + (1-G)(G^N) + …
This is a sum in the form of A + AB + AB^2 +  … + AB^N where B < 1.
When N goes to infinity the sum goes to A/(1 – B). Filling in A=(1-G) and B=G results in (1-G)/(1-G) = 1.

For girls we get (1-G)G + 2(1-G)(G^2) + … + N(1-G)(G^N) + … = (1-G) x (G + 2G^2 + 3G^3 + … + NG^N + …).
Here we have the sum of n*X^n which results in x/(x-1)^2.
The average number of girls is (1-G) (G / (G -1)^2) = G(1-G)/(G-1)(G-1) = -G/(G-1) = G/(1-G)

The average number of kids is 1 + G/(1-G) = 1/(1-G)

The percentage of girls is (G/(1-G)) / (1/(1-G)) = G(1-G)/(1-G) = G.

So the ratio of boys to girls will stay the same.