Infinite Sequences (The List of Numbers)

The Key Question

Does the list of numbers eventually settle down and approach one specific value?

The Test: Find the Limit

We check lim (n→∞) aₙ.

  • If the limit is a finite number (L), it CONVERGES to L.
  • If the limit is ∞, -∞, or does not exist, it DIVERGES.

Quick Limit Rules (for fractions)

  • Degree Top < Degree Bottom → Limit is 0.
  • Degree Top = Degree Bottom → Limit is Ratio of Coefficients.
  • Degree Top > Degree Bottom → DIVERGES (goes to ∞ or -∞).

Infinite Series (The Sum of the List)

The Key Question

Does adding up all the numbers in the list give you a single, finite sum?

CRITICAL First Step: The Divergence Test

First, find the limit of the terms in the list: lim (n→∞) aₙ.

  • If the limit is NOT ZERO, the series DIVERGES. STOP HERE! The sum cannot be finite if you keep adding non-zero numbers.
  • If the limit IS ZERO, the series *might* converge. You must test further.

Common Series Tests (if lim aₙ = 0)

Geometric Series: a + ar + ar² + ...

  • If |r| < 1, it CONVERGES to the sum S = a₁ / (1 - r).
  • If |r| ≥ 1, it DIVERGES.

P-Series: Σ (1/nᵖ)

  • If p > 1, it CONVERGES.
  • If p ≤ 1, it DIVERGES.