Calculus
Lesson
48
Sequences
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A sequence is just a list of numbers, separated by commas!
Examples (students solve for next term and an for each one):
The limit of a sequence is 
Examples:
Properties: add, const. mult., mult, divide.
no limit
take limit of top and bottom separately: 
Order of growth: 
demonstrate a couple of these
Squeeze Theorem: 
But we also know 
so
.
Example: 
Absolute limits: if 
then 
from the squeeze
theorem.
Monotonic Definition: 
or
Bounded Definition:
an
is "bounded above" and M is "upper
bound"
an
is "bounded below" and N is "lower bound"
an
is "bounded"
IMPORTANT: If a sequence is bounded and monotonic, then it has a limit (converges)!!!!!
Examples: Use a, b, and c from previous page.
an is not monotonic no guarantee!!!!
bn is monotonic increasing and bounded by 0 and 2 limit exists!!!
cn is not monotonic, but it is bounded by 0 and 2 no guarantee!!!!