Rosetta Stone · 1 of 30
Summation (capital sigma)
Operator: it does something
In plain English
Add up a list of terms. The bottom of the sign says where the counter starts, the top says where it stops, and the expression to the right is what gets added on each pass. means . When the limits are missing, as in , the sum runs over every item in an obvious set: all outcomes, all tokens, all particles.
As code
let sum = 0; // start at the additive identity
for (let n = 0; n <= 4; n++) {
sum += 3 * n; // the term to the right of Σ
}
// sum === 30Where the analogy bends: Exact for finite sums. Infinite sums (upper limit ∞) are limits, not loops that finish; they may or may not converge.
See it
What it means, field by field
Finite or infinite series; the index variable (n, i, k) is a dummy name with no meaning outside the sum.
Total energy, momentum or charge over all particles; sums over states in statistical mechanics (the partition function Z = Σ e^{−E/kT}).
Totals over a population, over sites in a sequence, or over species in a diversity index.
Loss averaged over a batch; attention as a weighted sum of value vectors; the dot product is a sum of products.
A sum is only as meaningful as the terms you chose to include. Ask what the index runs over.
In a research paper
For every possible outcome i, multiply its probability by the log of that probability, add them all up, and flip the sign.
Common misreadings
Σ as a variable named "sigma". As an operator it is an instruction, not a quantity; lowercase σ is a different symbol entirely.
Assuming the index starts at 1. Always read the bottom limit: n = 0 and n = 1 give different totals.
Treating an infinite sum as if it obviously has a finite value. Convergence must be checked or established.