Chapter 9
Sequences and Series
The Fibonacci sequence is a series of integers mentioned in a book by
Leonardo of Pisa (Fibonacci) in 1202 as the answer to an ancient
arithmetic problem. The series begins with zero, and naturally
progresses to one. The series becomes 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,
and so on into infinity. In the Fibonacci sequence, each number is
added to the previous to make the next.
9.1
Overview
Example 1
Example 2
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Sequences and Series



A series is the sum of the terms of a sequence.
Finite sequences and series have defined first and
last terms
Infinite sequences and series continue indefinitely.
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9.2
Sequences
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Practice with examples: a) b) c) p.527
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Definitions:
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Examples:
Bounded and
monotonic.
Limit is 1
Bounded but
non-monotonic.
Limit does not exist
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Definitions: Arithmetic and Geometric sequences
An arithmetic sequence goes from one term to the next
by adding (or subtracting) the same value.
Examples:
2, 5, 8, 11, 14,... (add 3 at each step)
7, 3, –1, –5,... (subtract 4 at each step)
A geometric sequence goes from one term to the next by
multiplying (or dividing) by the same value.
Examples:
1, 2, 4, 8, 16,... (multiply by 2 at each step)
81, 27, 9, 3, 1, 1/3,... (divide by 3 at each step)
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Practice:
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Practice:
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Practice:
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9.3
Infinite Series
Recall:

A series is the sum of the terms of a sequence.

Finite sequences and series have defined first
and last terms.

Infinite sequences and series continue
indefinitely.
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Geometric series

Constant ratio between successive terms.
Example:

Geometric series are used throughout mathematics, and they have
important applications in physics, engineering, biology, economics,
computer science, and finance.

Common ratio: the ratio of successive terms in the series
Example:

The behavior of the terms depends on the common ratio r.
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
If r is between −1 and +1, the terms of the series become smaller and
smaller, and the series converges to a sum.

If r is greater than one or less than minus one the terms of the series
become larger and larger in magnitude. The sum of the terms also gets
larger and larger, and the series has no sum. The series diverges.

If r is equal to one, all of the terms of the series are the same. The series
diverges.

If r is minus one the terms take two values alternately (e.g. 2, −2, 2, −2,
2,... ). The sum of the terms oscillates between two values (e.g. 2, 0, 2, 0,
2,... ). This is a different type of divergence and again the series has no
sum. (example Grandi's series: 1 − 1 + 1 − 1 + ···).
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Formula:
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Practice:
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Answers:
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Practice 2:
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Answers:
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Application of geometric series: Repeating decimals
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Telescoping series
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A term will cancel with a term that is farther down the list.
It’s not always obvious if a series is telescoping or not until you try to get
the partial sums and then see if they are in fact telescoping.
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9.4 – 9.5 – 9.6
Convergence Tests for Infinite Series
Alternating Series
Definition of Convergence for an infinite series:


Let
a
n 1

n
be an infinite series of positive terms.
The series converges if and only if the sequence of partial
sums,

S n  a1  a2  a3  a , converges. This means: lim S n   a n
n 
n 1
Divergence Test:

If lim a  0, the series
n
n 
a
n 1

Example: The series 
n 1
However,

n
n 1
2
n
diverges.
is divergent since lim
n 
n
n 1
2
 lim
n 
1
1 1
1
n2
lim a n  0 does not imply convergence!
n 
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Geometric Series:
a  ar  ar 2    ar n 1   converges for  1  r  1
a
If the series converges, the sum of the series is:
1 r
The Geometric Series:

7
Example: The series  5 
n 1  8 
n
with a  a1 
35
7
and r 
converges .
8
8
The sum of the series is 35.
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p-Series:

The Series:
1

p
n
n 1
and diverges for
(called a p-series) converges for
p 1
p 1

Example: The series 
n 1
1
n1.001

is convergent. The series
1
is divergent.

n
n 1
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Integral test:
If f is a continuous, positive, decreasing function on
[1, ) with f (n)  a n

then the series
a
n 1

n
converges if and only if the improper integral
 f ( x)dx
converges.
1

Example: Try the series:
1

3
n 1 n
Note: in general for a series of the form:
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Comparison test:


If the series
n 1

(a)
If
If

and
b
n
n 1
is convergent and
b
is divergent and
n 1

n
b
n 1

(b)
a
n
n
are two series with positive terms, then:
a n  bn
an  bn

for all n, then
a
n 1

for all n, then
a
n 1
n
n
converges.
diverges.
(smaller than convergent is convergent)
(bigger than divergent is divergent)



3n
3n
1


3



2
2
n2 n  2
n2 n
n2 n
Examples:
which is a divergent harmonic series. Since the
original series is larger by comparison, it is
divergent.


5n
5n 5  1
 3   2

3
2
2 n 1 n
2
n

n

1
n 1
n 1 2n
which is a convergent p-series. Since the original
series is smaller by comparison, it is convergent.
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Limit Comparison test:


If the the series
a
n 1
where
0c
n
and
b
n 1
n
are two series with positive terms, and if lim
n 
an
c
bn
then either both series converge or both series diverge.
Useful trick: To obtain a series for comparison, omit lower order terms in the numerator and the denominator
and then simplify.



n
n
1
Examples: For the series  2
compare to  2   3 which is a convergent p-series.
n 1 n  n  3
n 1 n
n 1 n 2

For the series
n  n
3
n 1
n
 n2
 

  which is a divergent geometric


n
compare to
n 1 3
n 1  3 

n

n
series.
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Alternating Series test:

If the alternating series   1n 1 bn  b1  b2  b3  b4  b5  b6   satisfies:
n 1
bn  0 then the series converges.
bn  bn1 and lim
n 
Definition: Absolute convergence means that the series converges without alternating (all signs
and terms are positive).

Example:
The series 
n 0
 1n
n 1
is convergent but not absolutely convergent.
(1) n
Alternating p-series 
converges for p > 0.
p
n
n 1


Example: The series

n 1
(1) n
n
(

1
)
and the Alternating Harmonic series
are convergent.

n
n
n 1

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Ratio test:

(a)
a
If lim n1  1 then the series  a n converges;
n a
n 1
n
(b)
If lim
(c)
Otherwise, you must use a different test for convergence.
an1
 1 the series diverges.
n a
n
If this limit is 1, the test is inconclusive and a different test is required.
Specifically, the Ratio Test does not work for p-series.
Example:
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Useful procedure:
Apply the following steps when testing for convergence:
1.
Does the nth term approach zero as n approaches infinity? If not, the
Divergence Test implies the series diverges.
2.
Is the series one of the special types - geometric, telescoping, p-series,
alternating series?
3.
Can the integral test, ratio test, or root test be applied?
4.
Can the series be compared in a useful way to one of the special types?
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Summary
of all tests:
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