Concise Summary of Power Series and Geometric Series in AP Calculus BC

Objectives

  • Find the sum of Geometric series.
  • Represent functions by series.

Sum of Finite Geometric Series

  • Formula:
    S = a + ar + ar² + … + ar^(n-1)
    Compact Form: S = ∑(k=1 to n) ar^(k-1)
  • Simplified: S = a(1 - r^n) / (1 - r)
    where r ≠ 1.

Sum of Infinite Geometric Series

  • Formula:
    S_n = ∑(n=1 to ∞) ar^n
  • Converges if |r| < 1: S = a / (1 - r)
  • Diverges if |r| ≥ 1.

Examples of Convergence

  1. Example: Does the series 3/10 + 3/100 + 3/1000 + … converge?
  2. Check convergence or divergence:
    • a) ∑(n=1 to ∞) 3(1/2)^(n-1)
    • b) 1 - 1/2 + 1/4 - 1/8 + …
    • c) ∑(k=0 to ∞) (3/5)^k
    • d) π/2 + π/4 + π/8 + …

Definition of Power Series

  • A power series centered at x = 0:
    ∑(n=0 to ∞) x^n = 1 / (1 - x)
    if -1 < x < 1.

Finding Power Series

  1. Example: Power series for 1 / (1 - 2x).
  2. Example: Finding the power series for 1 / (1 - x)² by differentiation.
  3. Integration of power series is also key.
  4. Example: Power series to represent ln(1 + x).