Power Series and Convergence

Definition of Power Series

• A power series is an infinite series of the form ( \sum a_n (x - c)^n ) where:
  • ( x ) is a variable.
  • ( c ) is a constant, known as the center of the power series.
  • If ( c = 0 ), the power series is centered at zero.

Domain of a Power Series

• The domain of a function ( f(x) ) represented by a power series is the set of all ( x ) values for which the series converges.
• A power series always converges at its center ( c ).
• The domain can be:
  • A single point (just ( c )).
  • An interval centered around ( c ).
  • All real numbers.

Radius of Convergence

• Denoted as ( R ).
• A series converges if (|x-c| < R) and diverges otherwise.
• Check endpoints separately to determine if they should be included in the interval of convergence.

Finding the Interval of Convergence

• The challenge is to determine when a power series converges.
• Use the Ratio Test:
  • Converges if the limit ( \lim_{{n \to \infty}} \left| \frac{a_{n+1}}{a_n} \right| < 1 ).
  • Apply to find when the ratio involving ( x ) is less than 1.

Example Investigation

• Investigate the graph of a specific power series to determine convergence:
  • Graph converges between ( x = -1 ) and ( x = 1 ).
  • ( x = 1 ) diverges (unbounded as ( n \to \infty )).
  • ( x = -1 ) converges (values stabilize).

Algebraic Analysis

• Use the Ratio Test for convergence:
  • Reduce the expression to find ( \lim_{{n \to \infty}} \frac{\sqrt{n}}{\sqrt{n+1}} ), leading to a limit of 1.
  • Radius of convergence is 1, so the interval of convergence is ( |x| < 1 ).

Endpoint Analysis

• Negative Endpoint ( x = -1 ):
  • Series ( \sum \frac{(-1)^n}{\sqrt{n}} ) converges by the Alternating Series Test:
    • Limit of nth term is 0 and terms decrease.
  • Include ( -1 ) in the interval.
• Positive Endpoint ( x = 1 ):
  • Series ( \sum \frac{1}{\sqrt{n}} ) diverges as a p-series with ( p = \frac{1}{2} < 1 ).
  • Do not include 1 in the interval.

Conclusion

• The domain of convergence is ( [-1, 1) ).
• Use either set notation or interval notation to express the domain.