Series and Sigma Notation

A sequence is a list. A series is that list added up. The distinction matters because they ask different questions.

Sigma notation

The Greek capital sigma means add. The letter below is the counter, its starting value sits below and the last value sits above.

Arithmetic series

For n terms, Sₙ = n(a₁ + aₙ) ÷ 2. It is the number of terms times the average of the first and last.

Why that works

Pair the first with the last, the second with the second-last, and so on. Every pair has the same total.

Geometric series

For a common ratio r not equal to 1, Sₙ = a₁(1 − rⁿ) ÷ (1 − r).

Count the terms

A sum from k = 1 to 10 has 10 terms, but from k = 0 to 10 it has 11. Off-by-one errors live here.

A series is a sum, a sequence is a list

The sequence 2, 4, 6, 8 becomes the series 2 + 4 + 6 + 8 = 20. The distinction is genuine, and questions that ask for a term and questions that ask for a total need different formulas.

Sigma notation

The Σ symbol with an index below and a limit above means "add these up". It is compact and unambiguous, and it is the notation every later course uses. Expanding a small example by hand is the way to become comfortable with it.

The arithmetic sum formula

Sₙ = n(a₁ + aₙ)/2 — the number of terms times the average of the first and last. Pairing the first with the last, the second with the second-to-last, and so on gives the same total each time, which is where the formula comes from.

The geometric sum formula

Sₙ = a₁(1 − rⁿ)/(1 − r). It is derived by subtracting r times the sum from the sum, which collapses almost every term. That trick is worth seeing once, because the same idea appears repeatedly in later mathematics.