Transcript Angles, Degrees, and Special Triangles

Sequences & Series
MATH 109 - Precalculus
S. Rook
Overview
• Section 9.1 in the textbook:
– Infinite sequences
– Factorial notation
– Partial sums & summation notation
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Infinite Sequences
Infinite Sequences
• We have discussed finite (countable) lists of numbers
when constructing a table of values:
– Given a function f(x), pick values of x to get f(x)
– We do this about 2 or 3 times to get an idea what f(x) looks
like
– Represents only a subset of the values of f(x)
– i.e. a Finite Sequence
• Infinite Sequence: a function whose domain is the
natural numbers. The results that are generated
from a sequence are its terms
• There are many infinite sequences of interest to
mathematicians and scientists
– Prime numbers, Fibonacci numbers, etc.
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Terms of a Sequence
• The nth term of a sequence also called the
general term is usually written an = f(n)
• Given a natural number k such that 1 ≤ k ≤ n,
we can find the kth term of the sequence by
simply substituting
– i.e. ak = f(k)
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Alternating Sequences
• Alternating Sequence: a sequence in which
subsequent terms change from positive to
negative or vice versa
– Has a general term such as an = (-1)n + 1 · f(n)
• Substitute as before to evaluate a term
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Infinite Sequences (Example)
Ex 1: For each sequence, find the first three
terms and then the 10th term:
a) an  3n  4
n 1
b) bn   1 
n 1
n
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Recursive Sequences
• Recursive Sequence: a sequence defined in
terms of itself using previous terms
– Usually given at least the first term of the
sequence
– e.g. an + 1 = 5 + an; a1 = 2
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Recursive Sequences (Example)
Ex 2: Find the first three terms of the recursive
sequence:
a) a1  2; an  4an1  2
b) b1  1; bn  n  1 bn1
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Factorial Notation
Factorial Notation
• Suppose we were give the recursive sequence
an = n · an – 1; a1 = 1
n = 2:
n = 3:
n = 4:
:
:
a 2 = 2 · a1 = 2 · 1 = 2
a3 = 3 · a2 = 3 · (2 · 1) = 3 · 2 = 6
a4 = 4 · a3 = 4 · (3 · 2 · 1) = 4 · 6 = 24
an = n · (n – 1) · (n – 2) · … · 2 · 1
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Factorial Notation (Continued)
• an = n · (n – 1) · (n – 2) · … · 2 · 1 is used often enough
that it is given the special name factorial and written
as n!
n! means the product of n down to 1
3! = 3 · 2! = 3 · 2 · 1! = 3 · 2 · 1 = 6
1! AND 0! are both equivalent to 1
n! = n · (n – 1)!
• We can use factorials when performing Algebraic
operations
– By expanding the factorial into a product
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Factorial Notation (Example)
Ex 3: Evaluate the factorials by hand:
6!
a)
4!
8!
b)
5!2!

6k !
c)
6k  1!
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Partial Sums & Summation
Notation
Partial Sums
• We have seen how to generate successive terms
from the sequence an = f(n)
• Another important series concept is the summation
of these terms
• The summation through the nth term is called the nth
partial sum denoted Sn
S1 = a 1
S2 = a 2 + a 1
S3 = a 3 + a 2 + a 1
:
Sn = an + an-1 + … + a2 + a1
• Each of the nth partial sums forms a sequence
• Sn is also called a finite series
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Summation Notation
• A shorthand way to write the partial sum from
the mth term to the nth term where m ≤ n is
n
where ∑ means to
a

a

a



a

k
m
m 1
n
sum the elements from m
k m
to n of the sequence an
m is known as the lower limit (starting value) of the
summation (does not always have to start at 1)
n is known as the upper limit (ending value) of the
summation
k (in this case) is known as the index of summation
(other variables can be used as well)
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Summation Notation (Continued)
– The summation of ALL terms of an infinite sequence is
known as an infinite series denoted in summation
notation as 
 ai  a1  a2    ai  
i 1
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Summation Notation (Example)
Ex 4: Evaluate:
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4
a)
1
1
b)  
j 1
j 6 j
i
i 1
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c)   1
k 1
k 1

1
d)  x
x 1 10
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Summary
• After studying these slides, you should be able to:
– Calculate the terms of the following types of sequences:
• Infinite
• Alternating
• Recursive
– Understand factorial notation and be able to perform simple
calculations
– Evaluate partial sums and series of a sequence using
summation notation
• Additional Practice
– See the list of suggested problems for 9.1
• Next lesson
– Arithmetic Sequences & Partial Sums (Section 9.2)
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