Calculus II - Power Series

Introduction to Series

• Previously focused on series convergence.
• New focus: Specific kinds of series and their applications.

Power Series

• Definition: A series of the form ( \sum_{n=0}^{\infty} c_n (x - a)^n ) where ( a ) and ( c_n ) are constants.
• Key feature: Function of ( x ) unlike other series which involve only numbers._

Convergence of Power Series

• Convergence depends on the values of ( x ).
• A power series may converge for some ( x ) values and diverge for others.

Terminology

• Radius of Convergence (R): The value such that the series converges for (|x-a| < R) and diverges for (|x-a| > R).
• Series may or may not converge at (|x-a| = R).
• Interval of Convergence: All ( x ) values including endpoints where the series converges.

Concepts

• If ( R ) is the radius of convergence:
  • The series converges for ( a - R < x < a + R ).
  • The series diverges outside this interval.
  • Endpoint convergence must be checked separately.

Example Analysis

• Example 1: Given series ( \sum_{n=1}^{\infty} \left(-1\right)^n \frac{n}{4^n}(x+3)^n )

  • Determines convergence properties based on endpoints.

• Example 2: Series ( \sum_{n=1}^{\infty} \frac{2^n}{n}(4x-8)^n )

  • Analyzes radius and interval of convergence.

• Special Cases:

  • Example 3: Series ( \sum_{n=0}^{\infty} n!(2x+1)^n )
  • Example 4: Series ( \sum_{n=1}^{\infty} (x-6)^n n^n )
  • Summarized results:
    • If converges only at ( x = a ), then ( R = 0 ) and interval is ( x = a ).
    • If converges for all ( x ), ( R = \infty ) and interval is ( -\infty < x < \infty ).

Conclusion

• Understanding radius and interval of convergence is crucial.
• Different power series have unique convergence behaviors at endpoints.