A series is the sum of the terms of a sequence.
Arithmetic series sum: Sₙ = n(a₁ + aₙ)/2. Geometric series sum: Sₙ = a₁(1-rⁿ)/(1-r). Infinite geometric series converges to a/(1-r) when |r| < 1. Power series (like Taylor series) represent functions as infinite sums. Series are used in calculus, numerical analysis, signal processing, and approximating complex functions with simple polynomials.
Series were studied by ancient Greek mathematicians. Zeno's paradoxes involve infinite series. The sum of infinite geometric series was understood by Archimedes. Power series were developed by Newton, Leibniz, and Taylor in the 17th-18th centuries. The rigorous theory of convergent series was established by Cauchy and Abel in the 19th century.