How to understand the sum of all "fourier frequencies"?
Let me put a practical answer with the following Matlab / Octave Code :

L = 2*1000;     % signal sample count
n = 0:L-1;      % discrete-time index

Fs = 44100;     % sampling frequency
am = [1, 0.5, 2, 0.5, 0.3, 0.6, 0.1, 0.2];                % magnitudes of 8 components
fm = [882, 2646, 4410, 6615, 8820, 10000, 13230, 15876];  % frequencies

% Time domain signal : 
% x = a1*sin(2*pi*f1*n) + a2*sin(2*pi*f2*n) + ... + am*sin(2*pi*fm*n)
x = sin((2*pi/Fs)* n'*fm)*(am');    

% Frequency domain signal :
X = abs( fft( x, L ) );

% Display :
figure,subplot( 2,1,1 );
plot( n/Fs , x ); 
title(' signal as a sum of sinusoidal frequencies ' );
xlabel(' time [s]');
subplot( 2,1,2 );
plot(linspace( 0 , Fs/2 , L/2 ) , (2/L)*X(1:L/2) );
title(' Fourier (frequency) spectrum magnitude of the        signal' );
xlabel(' Frequencies from 0 to Fs/2 [Hz]' );
ylabel(' FFT Magnitude ');


The output will be :



The output speaks for itself. The upper plot is the time-domain signal as a superposition of (M=8) different sinusoidal signals each with a particular frequency and amplitude. The lower plot shows its frequency spectrum magnitude which displays a peak per frequency component in $x[n]$. The peak's amplitude is proportional to the time-domain component amplitude.