What is the difference $U - L$ between the following two fractions, where all variables are positive integers?
$$U = \frac{2{b^2}\left({b^2}{2^k}({2^k} - 1) + 2^{k+1} - 1\right)}{(2^{k+1} - 1)({b^2}{2^k} + 1)}$$
$$L = \frac{{b^2}\left({b^2}{2^{k+1}}({2^k} - 1) + 3(2^k) - 2\right)}{({2^{k+1}} - 1)({b^2}{2^k} + 1)}$$
Here is my attempt:
$$U - L = \frac{{2^{k+2}}{b^2} - 3({2^k}{b^2})}{({2^{k+1}} - 1)({b^2}{2^k} + 1)} = \frac{{2^k}{b^2}}{({2^{k+1}} - 1)({b^2}{2^k} + 1)}.$$
Is this correct? My motivation for this question was to show that $U > L$.
Yes, that is correct. I did the algebra myself and got the same result as you.
I also plotted the expressions into Geogebra as functions, replacing $b$ with $x$ to get a nice curve and making $k$ a number. I simplified a bit by using the common denominator to subtact the numerators. No matter the setting on $k$ I got the same graph for both. Here is one of the more interesting graphs:
It is not clear in this shot, but one of the graphs is a black solid line and the other is a red dashed line. The two lines clearly show the same graph.