Averaging out noise and SNR improvement Suppose I have a sequence of identical symbols carrying the same information at a sample rate equal to the symbol rate, why do we always talk about averaging them to average out noise, why not just  adding them all together without  averaging them? I suppose the improvement in SNR is going to be the same assuming that noise is uncorrelated from one sample or symbol to another, is this related to the limited dynamic range in DSP?
Yes, they both give you the same SNR gain. I suppose there might be a dynamic range issue, as you say, if a LOT of samples are added, but I think the main reason that the literature speaks of "averaging" instead of "adding" is because for systems that allow a variable integration rate, we can be assured that other parts of the system work the same, like AGC.