Series & Sigma Notation

A sequence lists terms; a series adds them.

Sigma notation compresses that instruction. The symbol Σ carries a starting index below, an ending index above, and a rule to the right.

Σ (from n=1 to 4) of 2n means 2+4+6+8 = 20.

Sum formulas save you from adding term by term:

Arithmetic: Sₙ = n(a₁ + aₙ)/2 — the count times the average of the first and last.
Geometric: Sₙ = a₁(1 − rⁿ)/(1 − r), for r ≠ 1.

Read the limits carefully. Starting at n = 0 rather than n = 1 changes the number of terms, which changes the answer.

A series is the sum of a sequence's terms, and sigma notation Σ_{k=m}^{n} a_k specifies the index variable, its bounds and the general term. The number of terms is n − m + 1, an off-by-one that is the most common source of error.

The arithmetic sum Sₙ = n(a₁ + aₙ)/2 = n[2a₁ + (n−1)d]/2 comes from pairing terms equidistant from the ends, each pair summing to a₁ + aₙ.

The geometric sum Sₙ = a₁(1 − rⁿ)/(1 − r) is derived by computing Sₙ − rSₙ, where all interior terms cancel — the telescoping technique.

Sigma notation is linear: Σ(ca_k + b_k) = cΣa_k + Σb_k, which allows a complicated sum to be split into standard pieces. Standard results worth knowing are Σ_{k=1}^{n} k = n(n+1)/2 and Σ_{k=1}^{n} k² = n(n+1)(2n+1)/6.

These become the Riemann sums of integral calculus, where the same notation describes areas as limits of sums.