Summation/Sigma notation There are lots of variants in the notation for summation. For example, $$\sum_{k=1}^{n} f(k), \qquad \sum_{p \text{ prime}} \frac{1}{p}, \qquad \sum_{\sigma \in S_n} (\operatorname{sgn} \sigma) a_{1 , \sigma(1)} \ldots a_{n , \sigma(n)}, \qquad \sum_{d \mid n} \mu(d).$$ 
What exactly is a summation? How do we define it? Is there a notation that generalizes all of the above, so that each  of the above summations is a variant of the general notation? Are there any books that discuss this matter? 
It seems that summation is a pretty self-evident concept, and I have yet to find a discussion of it in a textbook. 

Except for the case of the upper and lower limit, all the other summations are really just sums of the form $$\sum_{P(i)} f(i)$$
Where $P$ is a unary predicate in the "language of mathematics", and $f(i)$ is some function which returns a value that we can sum. In the case of the sum of prime reciprocals $P(i)$ states that $i$ is a prime number and $f(i)=\frac1i$. In the second sum, $P(i)$ was $i\in S_n$, and $f(i)$ was that summand term. And so on. 
