Sum
The sum is the result of adding two or more numbers together.
Formula
a + b = \text{sum}
Definition
The sum is the answer you get when you add numbers together; for a set of numbers, sigma notation compactly expresses this, $\sum_{k=1}^{n} a_k$. In analysis, infinite series extend this idea as limits of partial sums, $\sum_{k=1}^{\infty} a_k = \lim_{n \to \infty} S_n$ where $S_n = \sum_{k=1}^{n} a_k$; convergence of this limit is non-trivial and requires tests (ratio, root, comparison, integral), and divergent series can sometimes be assigned meaningful values via analytic continuation.
Example
In $6 + 4 = 10$, the sum is $10$; the sum of the first $5$ natural numbers, $1 + 2 + 3 + 4 + 5 = 15$, is written $\sum_{k=1}^{5} k = 15$ in sigma notation. Gauss as a child found the sum $1 + 2 + \ldots + 100 = 5{,}050$ by pairing: $(1+100) + (2+99) + \ldots = 50$ pairs each summing to $101$, giving the general formula $n(n+1)/2$. The geometric series $\sum_{k=0}^{\infty} r^k = 1/(1-r)$ for $|r| < 1$ gives $1 + 1/2 + 1/4 + \ldots = 2$ when $r = 1/2$, the rigorous resolution of Zeno's paradox.
Key Insight
Sum, total, and "all together" all describe the same thing: the result of addition. The Riemann zeta function $\zeta(s) = \sum_{n=1}^{\infty} n^{-s}$ is an infinite sum that encodes deep information about the distribution of prime numbers, and its analytic continuation to $s = -1$ gives the famous "sum" $1+2+3+\ldots = -1/12$ in a regularized sense, a striking example of how far the idea of a "sum" can be stretched beyond ordinary addition.