BC Calculus - Weebly
Transcript of BC Calculus - Weebly
![Page 1: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/1.jpg)
BC Calculus
Sec 11.2
Series
![Page 2: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/2.jpg)
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?
![Page 3: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/3.jpg)
Topics for today
Terminology & Notation & Properties
Test for Divergence (aka “nth term test”)
Geometric Series
Harmonic Series
Telescoping Test
![Page 4: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/4.jpg)
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
![Page 5: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/5.jpg)
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
![Page 6: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/6.jpg)
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
![Page 7: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/7.jpg)
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
![Page 8: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/8.jpg)
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)
![Page 9: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/9.jpg)
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.
![Page 10: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/10.jpg)
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
![Page 11: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/11.jpg)
![Page 12: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/12.jpg)
Test for Divergence
Aka
nth term test
![Page 13: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/13.jpg)
“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.
![Page 14: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/14.jpg)
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
![Page 15: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/15.jpg)
Geometric Series
![Page 16: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/16.jpg)
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
![Page 17: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/17.jpg)
Most common forms of
geometric series
a ar ar 2 ar3 ... ar n1 ... an1
r n1
or an0
r n
![Page 18: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/18.jpg)
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
![Page 19: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/19.jpg)
Let’s look at
convergence
![Page 20: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/20.jpg)
What if r =1?
...a a a a
In this case, the series diverges.
![Page 21: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/21.jpg)
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
![Page 22: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/22.jpg)
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
![Page 23: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/23.jpg)
A Geometric Series
Is convergent if 1r
and its sum is 1
1
a
r
If 1r , the geometric series is divergent
![Page 24: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/24.jpg)
Ex:
1
1
26
3
n
n
1
26,
3a r
6converges to 18
1 (2 / 3)
![Page 25: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/25.jpg)
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?
![Page 26: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/26.jpg)
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
![Page 27: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/27.jpg)
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
![Page 28: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/28.jpg)
You Try
2
13
6
7
n
nn
r
a
6
7
3 2
3 1 4
6 6 6
7 7 2401
![Page 29: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/29.jpg)
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
![Page 30: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/30.jpg)
You try.
What are a and r?
3
1
3
5
n
n
1
27
125
n
n
27 27
125 125r a
![Page 31: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/31.jpg)
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
![Page 32: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/32.jpg)
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
![Page 33: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/33.jpg)
Another disguise
1
cos3
n
n
1
1
2
n
n
1 1
2 2r a
![Page 34: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/34.jpg)
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.
![Page 35: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/35.jpg)
Harmonic Series
![Page 36: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/36.jpg)
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
![Page 37: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/37.jpg)
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.
![Page 38: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/38.jpg)
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.
![Page 39: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/39.jpg)
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
![Page 40: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/40.jpg)
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
![Page 41: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/41.jpg)
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
![Page 42: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/42.jpg)
How many terms are
needed?
???
1
1100
n n
4310 terms required
Pick any positive # and, eventually, the
Harmonic series will surpass it.
![Page 43: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/43.jpg)
Telescoping Test
(Last test of the day)
![Page 44: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/44.jpg)
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.
![Page 45: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/45.jpg)
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
![Page 46: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/46.jpg)
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
![Page 47: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/47.jpg)
Now, expand each
series
1
1
1
1
1
3
n
n
n
n
![Page 48: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/48.jpg)
Conclusion
21
2 5 converges to
4 3 6n n n
![Page 49: BC Calculus - Weebly](https://reader031.fdocuments.us/reader031/viewer/2022020912/6202b936806c58665957698d/html5/thumbnails/49.jpg)
Summary
Terminology
Series rules
Test for Divergence (nth term test)
Geometric Series
Harmonic Series
Telescoping Test