10. Section 10.1 Sequences

EQ – What does it mean for a sequence to converge? Section 10.1 Sequences EQ – What does it mean for a sequence to converge?

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.

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:

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.

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

Example: Plot Y=

WINDOW

GRAPH

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

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

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

Examples

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)

Examples Increasing Non-Monotonic Non-Monotonic

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.

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

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

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

Assignment Pg. 561: # 1-25 odd