Calculus 8.1

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8.1 Sequences
Craters of the Moon National Park, Idaho
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A sequence is a list of numbers written in an
explicit order.
nth term
Any real-valued function with domain a 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.
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A sequence is defined explicitly if there is a
formula that allows you to find individual terms
independently.
To find the 100th term, plug 100 in for n
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A sequence is defined recursively if there is a
formula that relates an to previous terms.
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.
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An arithmetic sequence has a common difference
between terms.
Example
Arithmetic sequences can be defined recursively
or explicitly
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An geometric sequence has a common ratio between
terms.
Example
Geometric sequences can be defined recursively
or explicitly
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If the second term of a geometric sequence is 6
and the fifth term is -48, find an explicit rule
for the nth term.
Example
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You can determine if a sequence converges by
finding the limit as n approaches infinity.
The sequence converges and its limit is 2.
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Absolute Value Theorem for Sequences
If the absolute values of the terms of a sequence
converge to zero, then the sequence converges to
zero.
Dont forget to change back to function mode when
you are done plotting sequences.
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