Lecture Notes: Representing Functions as Power Series

Key Concepts

• Power Series Representation: Expressing a function in terms of an infinite sum of powers of a variable.
• Geometric Series: A series with a constant ratio between successive terms.

Geometric Series Formula

• General Form: ( a \sum r^n = \frac{a}{1-r} )
• Convergence: Converges if ( |r| < 1 )

Power Series Representation

• A power series can be centered at any point ( x = c )
• Example: Represent ( f(x) = \frac{1}{1+x} ) as a power series.
  • Convert ( 1+x ) to ( 1 - (-x) )
  • Identify ( a = 1 ) and ( r = -x )
  • Power series: ( \sum_{n=0}^{\infty} (-1)^n x^n )_

Interval of Convergence

• Depends on the Common Ratio: ( |r| < 1 )
• Example: For ( f(x) = \frac{1}{1+x} ), interval of convergence is ( -1 < x < 1 )

Examples and Applications

Example 1: Function ( \frac{1}{x} )

• Convert expression: ( x = 1 + (x - 1) )
• Power Series: ( \sum_{n=0}^{\infty} (-1)^n (x-1)^n )
• Interval: ( 0 < x < 2 )_

Example 2: ( \frac{1}{1-x^3} )

• Power Series: ( \sum_{n=0}^{\infty} x^{3n} )
• Interval: ( -1 < x < 1 )_

Example 3: ( \frac{1}{3-x} )

• Center at 0 and 1
• Power Series: ( \sum_{n=0}^{\infty} \frac{x^n}{3^{n+1}} )
• Interval: ( -3 < x < 3 )_

Example 4: ( \frac{8}{2x-9} )

• Center at 3
• Power Series: ( \sum_{n=0}^{\infty} (-1) (\frac{2}{3})^{n} (x-3)^n )
• Interval: ( 1.5 < x < 4.5 )_

Example 5: ( \frac{x^3}{x+2} )

• Separation method: ( x^3 \times \frac{1}{x+2} )
• Power Series for ( \frac{1}{x+2} )
• Total Series: ( \sum_{n=0}^{\infty} (-1)^n x^{n+3} )
• Interval: ( -2 < x < 2 )_

Example 6: ( \frac{3}{x^2 + x - 2} )

• Partial Fraction Decomposition
• Power Series for each fraction
• Combined Series and Interval

Conclusion

• General Method: Identify the form ( \frac{a}{1-r} ), determine ( a ) and ( r ), write the series, find interval and radius of convergence.
• Applications: Use for approximating functions and analyzing intervals where series representation is valid.

Additional Tips

• Factor and simplify expressions for easier series representation.
• Check interval of convergence carefully and ensure correct endpoints.
• Use calculators to verify convergence and approximate values.