Can I compute Fourier series without looping through all frequencies?
Make sure you do not mix up FFT and DFT, they gives identical results, but are different.
DFT complexity is about $O(N^2)$ while FFT complexity is $\frac{34}{9}N\log_2(N)$. This makes a huge difference is computation time.

I do not know any faster algorithm for computing a complete spectrum than the FFT. If you need a subset of the spectrum (only few frequency bins), you could look at Goertzel Algorithm.   

According to wikipedia, as a rule of thumb, Gortzel algorithm will be faster than FFT if : 
$$ M &lt; \frac{5N_2}{6N}\log_2(N_2)$$

Where 


$M$ : Number of frequency bin you need
$N$ : Number of point in your dataset
$N_2$ : The first power of 2 bigger than N.