Power Series: Center, Radius, Interval of Convergence

Introduction to Power Series

• Definition: A power series is an infinite series with variable x raised to the n power.
• Example: One starting from zero to infinity.
• Centers:
  • Series centered at zero ( c = 0).
  • Series centered at a different value ( c ≠ 0).

Determining the Center of a Power Series

• Steps:
  • Rewrite the series in the form x - c or x + c.
  • Set the inside expression to zero and solve for x.
  • Examples provided:
    • x + 4 → centered at c = -4.
    • x → centered at c = 0.
    • 3x - 2 → centered at c = 2/3.

Radius and Interval of Convergence

• Using the ratio test: To determine the radius and interval of convergence.
• Ratio Test Formula:
  • Take the limit as n approaches infinity of u_(n+1) / u_n.
  • Interpret results:
    1. Limit = 0: Series converges for all x values.
    • Radius of convergence (r) = infinity.
    • Interval of convergence: (-∞, ∞).
    2. Limit = ∞: Series diverges for all x values except x = c.
    • Radius of convergence (r) = 0.
    • Interval of convergence: {c}.
    3. Limit < 1: Series converges for specific x values.
    • Use the form 1 / r |x - c| < 1.
    • Multiply both sides by r to get |x - c| < r.
    • Interval of convergence: c - r < x < c + r._

Example Problems

Example 1: x^n / n!

• Solution: Using Ratio Test
  • Limit = 0.
  • Radius of convergence = infinity.
  • Interval: (-∞, ∞).

Example 2: n! * x^n*

• Solution: Using Ratio Test
  • Limit = ∞.
  • Converges only when x = 0.
  • Interval: {0}.

Example 3: n! * (2x - 1)^n*

• Solution: Using Ratio Test
  • Limit = ∞.
  • Converges only when x = 0.5.
  • Interval: {0.5}.

Example 4: x^(2n) / (2n)!

• Solution: Using Ratio Test
  • Limit = 0.
  • Radius of convergence = infinity.
  • Interval: (-∞, ∞).

Example 5: √n * (x - 1)^n*

• Solution: Using Ratio Test
  • Evaluates to |x - 1| < 1 → 0 < x < 2.
  • Check endpoints x = 0 and x = 2 for convergence.
  • Interval of convergence: (0, 2).

Example 6: (x - 3)^n / n

• Solution: Using Ratio Test
  • Evaluates to |x - 3| < 1 → 2 < x < 4.
  • Check endpoints for convergence/divergence.
  • Interval of convergence: [2, 4).](streamdown:incomplete-link)

Example 7: (-1)^(n+1) * (x - 4)^n / (n*9^n)

• Solution: Using Ratio Test
  • Evaluates to |(x - 4)/9| < 1 → -5 < x < 13.
  • Check endpoints for convergence/divergence.
  • Interval of convergence: ( -5, 13].