Power Series Lecture Notes

Introduction to Power Series

• A power series is a series with a variable, like X, raised to some power.
• Power series are categorized into types based on their center:
  • Centered at zero
  • Centered at a constant C
• The primary focus is on understanding convergence of these series and the tests applicable to them.

Structure of Power Series

• General form: ( \sum_{n=0}^{\infty} a_n (x-c)^n )
• They can be expressed as a sum of terms involving constants ((a_n)) and powers of (x):
  • (a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \ldots)
• The variable (x) differentiates it from other series where terms only depend upon (n).

Convergence of Power Series

• Convergence can be determined using various tests, such as the Ratio Test:
  • Converges absolutely if the limit is less than 1.
  • Important concepts include the radius of convergence and endpoints.
• The power series converges in an interval centered around (c), known as the interval of convergence.

Types of Convergence

1. Convergence at the center only: Radius of convergence (R = 0).
2. Convergence for all (x): Radius of convergence (R = \infty).
3. Convergence within a specific radius: Check using tests like the Ratio Test.

Testing for Endpoints

• Each endpoint of the interval of convergence must be tested individually for convergence using known series tests.

Power Series Examples

• Example 1: (\sum_{n=0}^{\infty} x^n) is a geometric series with convergence interval ((-1, 1)).
• Example 2: If a series converges for all (x), the interval is ((-\infty, \infty)).
• Example 3: More complex series may converge differently; use derivative tests, p-series, etc., to determine behavior at endpoints.

Calculus of Power Series

Derivatives

• Differentiate term-by-term:
  • The series becomes ( \sum_{n=1}^{\infty} n a_n (x-c)^{n-1} )

Integrals

• Integrate term-by-term:
  • The series becomes ( \sum_{n=0}^{\infty} \frac{a_n}{n+1} (x-c)^{n+1} + C )

Effects on Convergence

• Derivatives may lose convergence at endpoints, integrals may gain it.

Power Series Representation of Functions

• Find power series representations for functions, for example, represent ( \ln(1-x) ) as a power series on the interval ((-1, 1)).
• Use known series representations and integrate or differentiate to adjust them for new functions.

Summary

• Power series are a powerful tool for representing functions.
• Understanding the convergence and the tests applicable is crucial to working with them.
• Calculus can be applied to power series, impacting their convergence characteristics.