Key Concepts of Power Series

Power series are infinite series that express functions as sums of terms based on powers of ( (x - c) ). Understanding their convergence, radius, and representation helps in approximating functions and solving complex problems in Calculus II.

1. Definition of a power series

   • A power series is an infinite series of the form ( \sum_{n=0}^{\infty} a_n (x - c)^n ), where ( a_n ) are coefficients and ( c ) is the center of the series.
   • It represents a function as a sum of terms based on powers of ( (x - c) ).
   • Power series can converge for certain values of ( x ) and diverge for others.

2. Radius of convergence

   • The radius of convergence ( R ) determines the interval around the center ( c ) where the power series converges.
   • It can be found using the formula ( R = \frac{1}{\limsup_{n \to \infty} |a_n|^{1/n}} ).
   • If ( |x - c| < R ), the series converges; if ( |x - c| > R ), it diverges.

3. Interval of convergence

   • The interval of convergence is the set of all ( x ) values for which the power series converges.
   • It is typically expressed as ( (c - R, c + R) ) but may include endpoints depending on convergence tests.
   • To determine convergence at the endpoints, additional tests must be applied.

4. Taylor series

   • A Taylor series is a specific type of power series that represents a function as an infinite sum of its derivatives at a single point.
   • The general form is ( f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!} (x - c)^n ).
   • It provides a powerful tool for approximating functions near the point ( c ).

5. Maclaurin series

   • A Maclaurin series is a special case of the Taylor series centered at ( c = 0 ).
   • The general form is ( f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n ).
   • It is particularly useful for functions that are easily evaluated at zero.

6. Geometric series

   • A geometric series is a power series of the form ( \sum_{n=0}^{\infty} ar^n ) where ( a ) is the first term and ( r ) is the common ratio.
   • It converges if ( |r| < 1 ) and diverges if ( |r| \geq 1 ).
   • The sum of a convergent geometric series can be expressed as ( \frac{a}{1 - r} ).

7. Differentiation of power series

   • Power series can be differentiated term by term within their interval of convergence.
   • The derivative of a power series ( \sum_{n=0}^{\infty} a_n (x - c)^n ) is ( \sum_{n=1}^{\infty} n a_n (x - c)^{n-1} ).
   • This property allows for the manipulation and analysis of functions represented by power series.

8. Integration of power series

   • Power series can also be integrated term by term within their interval of convergence.
   • The integral of a power series ( \sum_{n=0}^{\infty} a_n (x - c)^n ) is ( \sum_{n=0}^{\infty} \frac{a_n}{n+1} (x - c)^{n+1} + C ).
   • This facilitates finding antiderivatives of functions represented by power series.

9. Representation of functions as power series

   • Many functions can be expressed as power series, allowing for easier computation and analysis.
   • Common functions represented include exponential, trigonometric, and logarithmic functions.
   • This representation is crucial for approximating functions and solving differential equations.

10. Convergence tests for power series

    • Common tests include the Ratio Test and the Root Test, which help determine the radius and interval of convergence.
    • The Ratio Test involves evaluating ( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| ).
    • The Root Test involves evaluating ( \lim_{n \to \infty} \sqrt[n]{|a_n|} ) to assess convergence.