Short answer: not quite.
The practical point of view on quantum field theory today is that you won't need the higher energy corrections as long as you keep yourself at low energies. In this sense, in practice, we should never need more than $N$ terms as long as the energies are low enough. The reason is that the corrections enter the equations in the form
$$\text{stuff we know} + c_n\left(\frac{E}{\Lambda_{\text{UV}}}\right)^n \cdot \text{something new} + \cdots,$$
where $E$ is the energy we are looking at and $\Lambda_{\text{UV}}$ (called the ultraviolet cutoff) is a very large energy at which the theory breaks down. $c_n$ is the new parameter we need to measure (some fundamental constant, like the electron charge for example).
If $E$ is very small compared to $\Lambda_{\text{UV}}$, we don't need some corrections because $E/\Lambda_{\text{UV}} \ll 1$ and they are very small compared to what we see. However, as $E$ gets larger, $E/\Lambda_{\text{UV}}$ gets closer to one and all the corrections start getting more important.
This is the trouble. General relativity has infinitely many of these corrections. Hence, if we get to sufficiently high energies (near the Planck scale), all of them are important. At this point, the quantum field theory is not quite consistent anymore and we do need a new theory of quantum gravity.
Think of it as describing water as a fluid. It works well for the ocean, it works well for a pool, it works well for droplets, although each case has a few more important aspects than others (surface tension is very important for droplets, maybe not as important for the ocean). However, if you zoom in close enough, you find out water is actually a molecule. At this point, the fluid description simply does not make sense, no matter how much information you have about the "fluid". You need a new description that can actually deal with molecules.
