1. Binomial Theorem

2. Go to slide 11

3. Introduction

4. Notice

5. Using Sigma Notation

6. Example

7. The Special Case ( 1 + x ) n

8. Example (1 + x ) 6

9. What about when n is not a natural number?

10. The Binomial Series

11. Let f(x) = ( 1 + x ) n, n ЄR-(NU{0}) Let’s find the Maclaurin series for f f (k) (0)f (k) (x)k 1( 1 + x ) n 0 nn( 1 + x ) n-1 1 n(n-1)n(n-1)( 1 + x ) n-2 2 n(n-1)(n-2)n(n-1)(n-2)( 1 + x ) n-3 3 n(n-1)(n-2)(n-3)n(n-1)(n-2)(n-3)( 1 + x ) n-4 4 n(n-1)(n-2)(n-3)….[n-(k-1)] n(n-1)(n-2)…..….[n-(k-1)] ( 1 + x ) n-k k

12. Thus the Maclaurin series for (1+x) n

13. Thus the Maclaurin series for f

14. Finding the interval of convergence of this series:

15. Example Expand and then use the expansion to obtain a rough estimation of 1/√2

16. Solution

18. Approximating 1/√2

19. Homework