Skip to content
The Lumenary
Grades 9–12AnalysisPlanned

Sequences and series

Arithmetic and geometric sequences, summation notation, sums of finite series, recursive definitions, and an informal look at convergence of infinite geometric series.

math.analysis.sequences-series

Show everything needed to reach this topic

The lesson for this topic has not been written yet. The outline below is part of the curriculum map and has not been reviewed.

Builds on

Functions and their graphs (precalculus)Proof: direct, contrapositive, contradiction and induction

Required prerequisites, all the way down: 24

You will learn to

  • Find formulas for the n-th term and the partial sums of arithmetic and geometric sequences, and prove them by induction.
  • Use recursive definitions and summation notation.
  • Explain why 1 + 1/2 + 1/4 + ⋯ = 2 and decide when an infinite geometric series converges.

Definition

  • A sequence of real numbers is a function from ℕ (or from the positive integers) to ℝ.
  • An arithmetic sequence has a constant difference between terms; a geometric sequence has a constant ratio.
  • An infinite series Σ aₙ converges to S if its partial sums approach S.

Theorem

  • For r ≠ 1, a + ar + ⋯ + arⁿ⁻¹ = a(1 − rⁿ)/(1 − r).
  • If |r| < 1 then Σ_{k≥0} ar^k = a/(1 − r); if |r| ≥ 1 and a ≠ 0, the series diverges.
  • The harmonic series 1 + 1/2 + 1/3 + ⋯ diverges, although its terms tend to 0.

Common misconceptions

  • Thinking a series must converge if its terms tend to zero.
  • Believing an infinite sum must be infinite.
  • Forgetting that 0.999… is a geometric series whose sum is exactly 1.

Leads to

Calculus of one variableRecurrences and generating functions

Optional discoveries