Sequences.ppt

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SEQUENCES
DEF
A sequence is a list of numbers in a given order
Example
first term
second term
n-th term
index
Example
Example
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SEQUENCES
DEF
A sequence is a list of numbers in a given order
Example
Example
Example
Find a formula for the general term of the
sequence
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SEQUENCES
Example
Find a formula for the general term of the
sequence
Example
Find a formula for the general term of the
sequence
the digit in the th decimal place of the number pi
Example
Find a formula for the general term of the
sequence
This sequence arose when the 13th-century Italian
mathematician known as Fibonacci
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SEQUENCES
Recursive Definitions
Example
Find a formula for the general term of the
sequence
This sequence arose when the 13th-century Italian
mathematician known as Fibonacci
Example
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SEQUENCES
PLOT THE SEQUENCES
Example
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SEQUENCES
LIMIT OF THE SEQUENCES
Example
Example
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SEQUENCES
Convergence or Divergence
Example
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SEQUENCES
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SEQUENCES
Example
Determine whether the sequence is convergent or
divergent.
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SEQUENCES
Example
Note
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SEQUENCES
THEOREM
continuous
convergent
convergent
Example
Find
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SEQUENCES
Factorial
NOTE
Example
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SEQUENCES
THEOREM
(SQUEEZE THEOREM FOR SEQUENCES)
Example
Find
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SEQUENCES
THEOREM
(SQUEEZE THEOREM FOR SEQUENCES)
Example
Find
where
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SEQUENCES
Example
For what values of r is the sequence convergent?
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SEQUENCES
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SEQUENCES
DEFINITION
DEFINITION
bounded from above
bounded from below
Upper bound
Lower bound
If m is a lower bound but no number greater than
m is a lower bound then m is the greatest lower
bound
If M is an upper bound but no number less than M
is an upper bound then M is the least upper bound.
Example
Is bounded below
Example
greatest upper bound ??
Is bounded above by any number greater than one
If is bounded from above and below,
If is not bounded
we say that
bounded
unbounded
Least upper bound
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SEQUENCES
If is bounded from above and below,
If is not bounded
we say that
unbounded
bounded
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SEQUENCES
DEFINITION
non-decreasing
DEFINITION
non-increasing
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SEQUENCES
DEFINITION
non-decreasing
DEFINITION
non-increasing
DEFINITION
monotonic
if it is either nonincreasing or nondecreasing.
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SEQUENCES
non-increasing
DEFINITION
DEFINITION
non-decreasing
Example
Is the sequence increasing or decreasing
Example12
Is the sequence increasing or decreasing
2-solutions
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SEQUENCES
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SEQUENCES
How to find a limit of a sequence
THEOREM
continuous
convergent
convergent
THEOREM
(SQUEEZE THEOREM)
THEOREM
Every bounded, monotonic sequence is convergent
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SEQUENCES
How to find a limit of a sequence
(IF you can) use Math-101 to find the limit.
Use other prop. To find the limit
squeeze
Example
Example
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SEQUENCES
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SEQUENCES
TERM-082
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SEQUENCES
TERM-082
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SEQUENCES
TERM-092
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SEQUENCES
TERM-092
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