Even with infinite resources, expansion cannot overcome a continuous growth rate, in the long term. With infinite resources, and being able to move anywhere at the speed of light, the volume of space that would could occupy would be limited to a sphere x light years in radius, growing geometrically with time. Meanwhile, our population grows at a rate of n^x, exponential with time, which for any constant n > 1, will eventually overcome the geometric term.
Say for example all you need for a human is 1m^3 of space, then if we had infinite energy, and could move at will at the speed of light, and live anywhere, even deep space, and maintained our current growth rate of 1.1%/year, then we would run out of space when:
volume of sphere x light years radius = total volume of humans after x years of growth
4/3*pi*(299792458 x)^3 = 6.97*10^9 * 1.011^x
I don't this this has a closed form solution in algebra, so just approximating it: After somewhere between 5750 and 5800 years at our current growth rate, even with infinite energy, and the ability to travel at the speed of light, and nothing needed other than space to put our own bodies, we'd run out of space. It would be a 5800 light year radius ball of solid humans. Nothing beats exponential growth in the long term.
And excepting ftl travel, that's as overoptimistic as things can possibly be. We'd have has to use up all of the mass of the Earth, or the Sun, or all of the matter in the volume of space available to us long before that, just to turn into more humans, to maintain that growth rate. And if we had the ability to make more mass (we're assuming infinite available energy after all), we'd collapse into a black hole from our own mass long before we reached the above point. And more realistic scenarios can only be more limited than that.
Long term, the only solution is zero population growth, or at least a continuously decreasing growth rate, any constant exponential growth rate will eventually overcome anything you can throw at it.
