MAT 4830 Numerical Analysis Binomial Coefficients and Combinatorial
MAT 4830 Numerical Analysis Binomial Coefficients and Combinatorial Identities http: //myhome. spu. edu/lauw
Goals l l l Binomial Theorem Binomial Coefficients Combinatorial Identities Review shifting indices Review Induction
Take Home Exam l l Need Binomial Coefficients and Binomial Theorem for the second problem. No References besides your numerical and programming textbooks. The following serves as a review of the general subject. We will not go into all the details.
Binomial Expansion
Binomial Theorem
Useful Formulas for Binomial Coefficients
Pascal’s Identity
Pascal’s Identity Proof: Analysis
Binomial Theorem Combinatorial Proof: Analysis
Example 1 l
Example 2 Proof: Analysis
Example 3 (a) Proof: Analysis 1. Induction 2. Can be done without induction, but need to take care special cases.
Example 3 (b) Solution: Analysis
Binomial Theorem Induction Proof: Need some preparations Analysis
Binomial Theorem Proof: Analysis
3. Recall: Index Shifting for Summations (Use this if…)
Index Shifting l Sigma representation of a summation is not unique
Index Shifting Rules
Index Shifting Rules increase the i in the summation by 1 decrease the index by 1
Index Shifting Rules decrease the i in the summation by 1 increase the index by 1
- Slides: 20