Sigma Notation & Series
“Sigma is a one-letter command: 'add them all in order'.”
The formula
Σₖ₌₁ⁿ k = n(n+1)/2How to read it: adding every whole number from 1 up to n gives n(n+1)/2
- Σ
- — sigma — the symbol meaning 'add them all up'
- k
- — the running variable (it steps through 1, 2, 3, …)
- n
- — the upper limit — how far the sum runs
The hook
You don't have to add 1+2+3+…+100 one at a time — sigma notation and a single formula finish it in one line.
In plain words
Σ is a command: 'plug in from the bottom value up to the top value and add them all'. Σₖ₌₁ⁿ k is the sum from 1 to n.
The intuition
Lay 1-to-n forwards and backwards in two rows; every paired sum is (n+1). There are n such pairs, so twice the total is n(n+1), giving a sum of n(n+1)/2. That overlap-and-pair picture IS the formula.
How it's built
Below the Σ, k=1 is the start; above it, n is the end; the expression after is 'what to add'. Three sums are staples — Σk=n(n+1)/2, Σk²=n(n+1)(2n+1)/6, Σc=cn (a constant added n times).
Example
From 1 to 10: Σₖ₌₁¹⁰ k = 10·11/2 = 55. Indeed (1+10)+(2+9)+… = 11, five pairs = 55. It checks out.
Common mistake
Σ(k+1) is NOT Σk + 1. The constant 1 is also added n times, so it's Σk + n. It's easy to forget that every term inside the sigma repeats n times.
Where it's used
Means and variances, sums of arithmetic and geometric series, repeated addition in computing, and the seed of integration (slicing area and adding) — everywhere you 'add up many things'.
Where it came from
The sum symbol Σ is the Greek letter sigma (the S of 'sum'); Euler's widespread use of it made it the standard.
Prerequisites
Quick check
What is Σₖ₌₁⁵ k?
- 10
- 15✓
- 25
- 30
Practice
Find Σₖ₌₁¹⁰⁰ k, the sum from 1 to 100.
Answer: 5050
- Formula n(n+1)/2 with n=100
- 100·101/2 = 10100/2 = 5050
Takeaway: Gauss's trick of pairing the ends.
Find Σₖ₌₁⁴ 3 (the constant 3 added four times).
Answer: 12
- Sum of a constant: Σc = cn
- 3·4 = 12
Takeaway: A constant is added once per term.
Find Σₖ₌₁⁵ k², i.e. 1²+2²+3²+4²+5².
Answer: 55
- Formula n(n+1)(2n+1)/6 with n=5
- 5·6·11/6 = 330/6 = 55
Takeaway: Sums of squares use their own formula.
Find the sum from 1 to 20, Σₖ₌₁²⁰ k.
Answer: undefined
- Plug in n=20
- 20·21/2 = 420/2 = 210
Takeaway: One formula, however many terms, in a single step.