1. 10. Section Sequences

2. EQ – What does it mean for a sequence to converge?
Section Sequences
EQ – What does it mean for a sequence to converge?

3. nth term A sequence is a list of numbers written in an explicit order.
Any real-valued function where the domain is the subset of the positive integers is a sequence.
If the domain is finite, then the sequence is a finite sequence.
In calculus, we will mostly be concerned with infinite sequences.

4. A sequence is defined explicitly if there is a formula that allows you to find individual terms independently.
Example:
To find the 100th term, plug 100 in for n:

5. A sequence is defined recursively if there is a formula that relates an to previous terms.
Example:
We find each term by looking at the term or terms before it:
You have to keep going this way until you get the term you need.

6. Sequence Graphing on the Ti
Change the graphing mode to “sequence” and “dot”

7. Example: Plot Y=

8. WINDOW

9. GRAPH

10. We can also look at the results in a table.
TBLSET
TABLE
Scroll down to see more values.

11. 3 Important Concepts for Sequences
Writing a formula for the “nth” or general
term
Determining if sequence is increasing,
decreasing, or non-monotonic (increasing
and decreasing)
3. Convergence/Divergence

12. Writing a formula for nth term
Find a pattern
Write the pattern in terms of n

13. Examples

14. Determining Increasing/Decreasing/ Non-Monotonic
Three ways:
Compare an with an+1 then an+2, then an+3
Look at graph
Use 1st derivative test (always positive –
increasing, always negative – decreasing)

15. Examples
Increasing
Non-Monotonic
Non-Monotonic

16. Convergence
You can determine if a sequence converges by finding the limit as n approaches infinity.
Does converge?
The sequence converges and its limit is 2.

17. Example...
Does the sequence 1, 1/2, 1/4, 1/8, 1/16, /2n converge?
Does the sequence 1, 4, 9, 16, .…n , converge?
Converge 2
Diverge

18. Example The sequence Converges to 0 Converges to 1 C. Converges to n
D. Converges to ln 2
E. Diverges

19. Another... The sequence A. Converges to 0 B. Converges to 1
C. Converges to -1
D. Converges to ln 2
E. Diverges

20. Assignment
Pg. 561: # 1-25 odd