Sequences and series

Builds on

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

Optional discoveries

• Real analysis Rigorous analysis makes 'approach' precise with ε–N definitions.
• Dynamical systems and chaos Iterating x ↦ r x (1 − x) gives sequences that converge, oscillate or behave chaotically depending on r.
• ← Fractions and decimals
• ← Real numbers and irrationality

Learning pathways

• Analysis, PDE and dynamics