Lecture 15

Learning objectives

After this class, you should be able to:

1. Use generating functions to find closed form solutions to simple recurrences.

Reading assignment

1. Sec 7.3, up to page 338 (G(z) for the Fibonacci numbers) and class notes.
2. Page 338.

Exercises and review questions

• Questions on current lecture's material
  1. Use generating functions to derive the closed for solution to the following recurrence: g0 = 0, gn = 2gn-1 + 2n .
  2. Use generating functions to derive the closed for solution to the following recurrence: g0 = 0, gn = gn-1 + n . (Hint: Use equation 7.18 and a closed form expression from table 335.)

• Questions on next lecture's material
  1. Write the fraction 1/[(1-z)(1+z)] as the sum of two fractions, each of whose denominator is a polynomial of degree 1. (A polynomial of degree 1 has the form a*z + b, where a and b are constants.
  2. Find the roots of the following polynomials, using the formula for the solution of a quadratic equation: (i) z2 - z - 1 and (ii) -z2 - z + 1.
  3. If a quadratic polynomial has roots 2 and 1, and the coefficient of its highest degree term is 5, then determine the polynomial.
  4. What is the limit, as z approaches 2, for the rational function (z-2)/(z2-4)?
  5. What is the limit, as z approaches infinity, for the polynomial z2 + 8z - 1?
  6. If the limit, as z approaches infinity, for a polynomial is 4, then determine the polynomial.
  7. If the limit, as z approaches infinity, for a polynomial is 0, then determine the polynomial.