1. Numerical Sequences

2. Why Sequences? Animation 1 Animation 2 Infinitely close and direction

3. Why do we need to approach to the limit point systematically?

4. The behavior of the function g(x)=1/x was previously discussed for values of x "close to zero", for values of x "very large but positive", and for values of x "very large but negative".

5. Support from Tables Verify these tables with the calculator

6. Challenge

7. The graph of a function f(x) is given but you don’t know anything else about the function other than the graph and values of the functions domain for points "close" to zero. Use the graph of the function y=f(x) to make conjectures about the following values Are these conjectures supported by the tables?

8. The Graph in Smaller Window Discuss the values below in both graphs

9. Definitionof Numerical Sequences Listing – Finite – Infinite Formulas (graph as listing and as functions) a. b. c. d.

10. Basic Functions vs. Basic Sequences Show graphing sequences Indicate how to start always at zero Sequence as a listing vs. sequence as a function

11. Key types of sequences Diverging to infinity Converging to zero

12. Sequences Diverging to Infinity Number Positive Infinitely Large “∞” Slider Normally a LARGE NUMBER Slider Normally a LARGE NUMBER Slider Normally a LARGE NUMBER Slider Normally a LARGE NUMBER

13. EXAMPLE A=800 When N=29 First term passing slider What is N when A=10 28 ? Show in calculator

14. Exercise For the sequence below find the number of terms less than or equal to each of the given values of A. Represent the results geometrically in one and two dimensions. For each A determine the value N satisfying the condition

15. Positive Sequences Converging to Zero Numbers Positive Infinitely Small “0 +” SLIDER ε (small) distance from zero SLIDER ε (small) distance from zero SLIDER ε (small) distance from zero SLIDER ε (small) distance from zero

16. EXAMPLE When N=1001 first term passing SLIDER

17. Show in calculator When N=1001 first term passing SLIDER What is N when ?

18. EXERCISE For the sequence find the number of terms less than or equal to the given values of. Start by construction the table of values of the sequence, and represent the results on the number line/graph. Finally support your conclusions algebraically.

19. Negatively Infinitely large sequences Negatively Infinitely small sequences Relationship between positively large and positively small Sequences converging to zero – Positive – Negative

20. Converging to Any Number Other Than Zero Generate examples of “7 + ”, “7 - ”, “7”

21. Verification Time Provide four examples of – “Infinity” – “0” – “0.2 +”

22. Operations with “ “