Calculus II: Power Series

Introduction

• Focus on power series, a type of series written as: [ \sum_{n=0}^{\infty} c_n (x-a)^n ] where (a) and (c_n) are numbers.
• Different from previous series as it involves variables and can be considered a function of (x)._

Convergence of Power Series

• A major question is whether the series converges, which depends on the values of (x).
• The convergence can differ for various values of (x).

Terminology

• Radius of Convergence (R):
  • The range (|x-a| < R) where the series converges.
  • Series diverges for (|x-a| > R).
  • Convergence at (|x-a| = R) is undetermined and doesn't affect (R).
• Interval of Convergence:
  • The set of all (x) values for which the series converges.
  • Includes the interval ((a-R, a+R)).
  • Ends determined by testing if series converges at (x = a-R) or (x = a+R).

Specifics on Convergence

• At (x = a), the power series always converges because it collapses to (c_0).

Examples and Summary

• Example 1: Explore radius and interval of convergence where endpoints don't converge.
• Example 2: Similar analysis; emphasis on checking endpoints.

Special Cases

• Example 3 & 4:

  • Only converges at (x = a): (R = 0), interval (x = a).
  • Converges for all (x): (R = \infty), interval ((-\infty, \infty)).

• Example 5: Another case for determining radius and interval of convergence.

Conclusion

• Understanding power series involves determining the convergence, radius, and interval of convergence.
• Testing endpoints of intervals is crucial for identifying the full range of convergence.

References:

• Paul's Online Notes on Power Series

Page Last Modified: 11/16/2022