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Sequences and Series Cheat Sheet

By Dr. Iris Vaughan, Mathematics Editor·Published 15 November 2025·Last reviewed 1 February 2026

Arithmetic and geometric sequence formulas, convergence tests for series, and the standard series (geometric, p-series, alternating) with convergence conditions.

Key Formulas
Arithmetic: aₙ=a₁+(n−1)d, Sₙ=n(a₁+aₙ)/2
Geometric: aₙ=a₁rⁿ⁻¹, Sₙ=a₁(1−rⁿ)/(1−r)
Infinite geometric: S=a₁/(1−r), |r|<1
p-series: Σ1/nᵖ converges iff p>1

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