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TRANSCRIPT
BC Calculus
Sec 11.2
Series
Sequence vs. Series
Sequence:
A list of terms
Series:
The SUM of a sequence
1 2, ,... ...na a a
1 2 ... na a a
Our primary concern for the next 2 weeks:
Does a given series (sum) converge or diverge?
If it converges, what does it converge to?
Topics for today
Terminology & Notation & Properties
Test for Divergence (aka “nth term test”)
Geometric Series
Harmonic Series
Telescoping Test
Sequence vs Series
Notation
1Given n na s a
n
The sum of the first 4
elements of sequence a.4
1 1 11
2 3 4s
4
1The 4th element of sequence
4a a
Summation Notation
1
n
n k
k
s a
k is a counter. It starts at the lower number, 1 in this
case, and goes to the upper number, n in this case.
They are similar to the bounds of an integral.
na
1k 2k 3k k n
3a2a1a
4
4
1
1 1 1 1 1
1 2 3 4k
sk
The letters k and n are arbitrary. They can be anything.
The starting and stopping values of our counter can be
anything.
1
n
j
j
a
8
3
m
m
a
1
n
n
a
0
n
n
a
Examples
Partial Sums, aka finite series
11 as
212 aas
3213 aaas
nn aaaas
generalIn
...
:
321
…
Infinite Series
Partial sums add a finite number of elements.
An infinite series adds an infinite number of elements.
1 2 1
1
n n n
n
a a a a a
Convergence
We are most interested in if a series (sum) converges
to a specific number or not
A partial sum (aka finite series) always converges.
An infinite sum may converge or diverge.
(similar to improper integrals)
Properties of Series
1 1
n n
n n
ca c a
1 1 1
n n n n
n n n
a b a b
1 1 1
b
n n n
n n n b
a a a
2 2
1 1
1 13 3
1 1n nn n
1 1 1
1 1 1 1
1 3 1 3n n nn n n n
10
1 1 11
sin sin sin
n n n
n n n
n n n
Constants
Sums and Differences
Piecewise
These properties can be used to make it easier to
determine the behavior of the series.
General Process
• Analyze problem and choose most appropriate test
• Do mechanics of the test
• Interpret results
• If necessary, start again with a different test
All series we study can be analyzed.
So, for every type of test we discuss:
• Must know when a test can and cannot be used
• Know how to actually do the test
• Know how to interpret results, what the test can and
cannot tell you and when it is inconclusive
Test for Divergence
Aka
nth term test
“Test for Divergence”
aka “nth term test”
If limn
an does not exist or lim
na
n 0
then the series an
n1
diverges.
When can you use it:
Always. Normally the first test you do.
If the individual terms of the SEQUENCE are not
approaching zero, the SERIES diverges. Period.
If the terms do approach zero, the test is
inconclusive. More tests are required.
Use and interpret the
nth term test
2
21
2
1
21
21
2
3
2
2
2
4 3
n
n
n
n
n n
n
n n
n
n
n n
n n
2
2
2lim 1 Sequence converges to nonzero #
3
Series diverges
n
n n
n
2 2
lim Sequence diverges
Series also diverges
n
n n
n
2lim 0 Sequences converges to 0
2
Series may converge or diverge
n
n
n n
2
2lim 0 Sequences converges to 0
4 3
Series may converge or diverge
n n n
Geometric Series
One type of series you
are familiar with is:
The Geometric series
3 6 12 24 48
1 2 3 43 3 2 3 2 3 2 3 2
1 3 2a r
1 1
1
1 1
3 2n n
n n
a r
Most common forms of
geometric series
a ar ar 2 ar3 ... ar n1 ... an1
r n1
or an0
r n
The key clue
If the base is a constant and n is in the
exponent, it is most likely geometric.
1
1
3
1
1 1
2 2
1 1 1
4 4 4
1 1
2 2
n
n
n
nn
n
n
r
r
r
Let’s look at
convergence
What if r =1?
...a a a a
In this case, the series diverges.
Now let’s consider a general geometric
series where r does not = 1
132 ... n
n ararararas
nn
n arararararrs 132 ...
Mult. by r
Now, subtract the 2 eq. n
nn ararss
n
n arars )1(
1
(1 ) 1
nn
n
a ra ars
r r
1
(1 )
n
n
a rs
r
lim n
nr
(1 )lim lim
(1 )
n
nn n
a rs
r
lim(1 )
nn
as
r
If –1 < r < 1, then as n increases w/out bound 0
A Geometric Series
Is convergent if 1r
and its sum is 1
1
a
r
If 1r , the geometric series is divergent
Ex:
1
1
26
3
n
n
1
26,
3a r
6converges to 18
1 (2 / 3)
Beware
1
1
140
2
n
n
1
140
2
n
n
1
1
140 20
2
1
2
a
r
1 40
1
2
a
r
What are a and r?
Non-obvious
Geometric Series
1
21
2
5
n
nn
The bases are constants,
The exponents have n’s,
It is geometric.
1
131
2
5 5
n
nn
1
1
1 2
125 5
n
n
1 2
125 5 a r
What you should do
1
21
2
5
n
nn
You do NOT need to rewrite
the series. That takes time and
is prone to error.
What is being raised to some power of n ?2
5
What is the initial value? 1 1
1 2 3
2 1 1
5 5 125
a
You Try
2
13
6
7
n
nn
r
a
6
7
3 2
3 1 4
6 6 6
7 7 2401
Another Geometric
Incognito
2
1
2
3
n
n
2
1
2
3
n
n
1
4
9
n
n
4 4
9 9r a
You try.
What are a and r?
3
1
3
5
n
n
1
27
125
n
n
27 27
125 125r a
A little more
complicated
2 1
1
1
2
n
n
2
1
1 1
2 2
n
n
1 1
4 8r a
1
1 1
2 4
n
n
You try.
What are a and r?
3 2
1
1
3
n
n
32
1
1 1
3 3
n
n
1 1
27 243r a
1
1 1
9 27
n
n
Another disguise
1
cos3
n
n
1
1
2
n
n
1 1
2 2r a
What is the common
thread?
1
21
2
5
n
nn
2
1
2
3
n
n
2 1
1
1
2
n
n
1
cos3
n
n
They all have n in the exponent and
Constants in the base. That is the key.
Harmonic Series
Definition
1
1 1 1 1Harmonic Series = 1 ...
2 3 4n n
The name “Harmonic series” comes from the
world of music and overtones, or harmonics.
The wavelengths of the overtones of a
vibrating string are
Source: Wikipedia.com
1 1 1, , ...
2 3 4
Does the harmonic series
converge or diverge?
1
1 1 1 1Harmonic Series = 1 ...
2 3 4n n
1limn n
Try the nth term test.
0
It is not a geometric series.
Try the geometric test.
We have to get creative.
1
1 1 1 1 1...
1 2 3 4n n
We’re going to do a comparison test.
If a sum is clearly less than a series that is
known to converge then
If a sum is clearly greater than a series that
is known to diverge then
The sum must also converge.
The sum must also diverge.
First, let’s group the terms.
1
1 1 1 1 1...
1 2 3 4n n
...16
1...
10
1
9
1
8
1
7
1
6
1
5
1
4
1
3
1
2
11
Now, this ...16
1...
10
1
9
1
8
1
7
1
6
1
5
1
4
1
3
1
2
11
Clearly exceeds:
1 1 1 1 1 1 11
2 4 4 8 8 8 8
1 1 1... ...
16 16 16
...2
1
2
1
2
1
2
11 Which equals:
Which clearly increases forever and diverges.
So, must diverge.
1
1
n n
So, the Harmonic
Series Diverges.
How fast?
1000
1
1
n n
1 million
1
1
n n
7.4851 1 1
12 3 1000
1 billion
1
1
n n
14.357
21
How many terms are
needed?
???
1
1100
n n
4310 terms required
Pick any positive # and, eventually, the
Harmonic series will surpass it.
Telescoping Test
(Last test of the day)
21
2
4 3n n n
Try the term test.thn
2
2lim 0
4 3n n n
Test is inconclusive.
It is not geometric.
It is not the harmonic series.
Telescoping Test
When to use it:
When the nth term test is inconclusive AND
You can rewrite the rule with partial fractions.
21 1
2
4 3 1 3n n
A B
n n n n
Doing the Telescoping
Test
21 1
2 1 1
4 3 1 3n nn n n n
1 1
1 1
1 3n nn n
Do the partial
fraction work.
Rewrite as two
series
21 1
2
4 3 1 3n n
A B
n n n n
Now, expand each
series
1
1
1
1
1
3
n
n
n
n
Conclusion
21
2 5 converges to
4 3 6n n n
Summary
Terminology
Series rules
Test for Divergence (nth term test)
Geometric Series
Harmonic Series
Telescoping Test