Section 9.2 – Series and Convergence
description
Transcript of Section 9.2 – Series and Convergence
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Section 9.2 – Series and Convergence
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Goals of Chapter 9
• Approximate Pi• Prove infinite series are another important
application of limits, derivatives, approximation, slope, and concavity of functions.
• Find challenging antiderivatives like • Lay the groundwork for future courses
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Summation NotationA compact notation (often called sigma notation) for sums
is the following:
1 21
n
i ni
a a a a
Upper Limit of Summation
General Term
Lower Limit of Summation
Index of Summation
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Examples
Evaluate: 3
2
1
4 1i
i
24 1 1 i=1
24 2 1 i=2
24 3 1 i=3
3 15 35
53
Series investigate the following:
2
1
4 1i
i
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Infinite Sum
1
1 What is the area of the square?
square unitCut the square in half and label
the area of one section.𝟏𝟐
Cut the unlabeled area in half and label the area of one
section.𝟏𝟒 Continue the process…
𝟏𝟖 𝟏
𝟏𝟔
𝟏𝟑𝟐 𝟏
𝟔𝟒
𝟏𝟏𝟐𝟖
Sum all of the areas:
1 1 1 1 1 1 12 4 8 16 32 64 128 ... 1
2...n
The general term is…1
Since the infinite sum represents the area of the square…
12
1n
n
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Infinite Series
An infinite series is an expression of the form
, or
The numbers are the terms of the series; is the nth term.
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Connecting Series and SequencesConsider the Series:
Find the sum of…The first term: The first 2 terms:The first 3 terms:The first 4 terms:
Consider the sequence of PARTIAL SUMS above:
The sequence of PARTIAL SUMS appear to converge to:
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Partial Sums of a SeriesThe partial sums of the series for a sequence:
of real numbers, each defined as a finite sum.
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Convergent or Divergent Series
If the sequence of partial sums has a limit as , we say the series converges to the sum , and we write
Otherwise, we say the series diverges.
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ExamplesInvestigate the partial sums of the sequences below to determine if the series converges or diverges. If it converges, state the limit.
1. 2. 3. 4. 5. 6.
¿𝟏−𝟏+𝟏−𝟏+𝟏−𝟏+…
¿𝟎 .𝟓+𝟎 .𝟎𝟓+𝟎 .𝟎𝟎𝟓+𝟎 .𝟎𝟎𝟎𝟓+…¿𝟏+𝟑+𝟓+𝟕+…¿𝟎 .𝟏+𝟎 .𝟐+𝟎 .𝟒+𝟎 .𝟖+…
¿𝟓𝟎+𝟐𝟓+𝟏𝟐 .𝟓+𝟔 .𝟐𝟓+…
𝑫𝒊𝒗𝒆𝒓𝒈𝒆𝒔¿𝟓𝟗
𝑫𝒊𝒗𝒆𝒓𝒈𝒆𝒔𝑫𝒊𝒗𝒆𝒓𝒈𝒆𝒔
¿𝟏𝟎𝟎
Why do 1, 3, 4 and 6 Diverge? The limit of the general term does not equal 0.
¿𝟑+𝟐 .𝟓+𝟐 .𝟑𝟑+𝟐 .𝟐𝟓+…𝑫𝒊𝒗𝒆𝒓𝒈𝒆𝒔
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The n-th Term TestIf , then the infinite series diverges.
OR
If the infinite series converges, then .
Is the converse of this statement true?If , does the infinite series always converge?
When determining if a
series converges, always use this
test first!
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The Converse of The n-th Term Test
Consider the two famous sequences below:
1
1 1 1 1: 1 ... ...2 3 4
1 1 1 1: 1 ... ( 1) ...2 3 4
n
HarmonicSeries n
AlternatingHarmonic Series n
For both series’, the . BUT do both series’ converge?Check a calculator program.
1lim 0n n
1 1lim( 1) 0n
n n
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The Converse of The n-th Term Test
11 11 1
: : ( 1)nn nn n
Harmonic AlternatingSeries Harmonic Series
The Alternating Harmonic Series appears to converge to ~0.69.
The Harmonic Series appears to diverge.
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The n-th Term TestIf , then the infinite series diverges.
OR
If the infinite series converges, then .
The converse of this statement is NOT true.If , the infinite series does not necessarily converge.
When determining if a
series converges, always use this
test first!
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The Harmonic Series DivergesProve the Harmonic Series diverges:
Compare the Series to the graph of .
……
Find the Left Hand Riemann Sum to approximate .
1 12 1
3 14
15 ...
The Left Hand Riemann Sum is equal to the Sum of the
Harmonic Series.
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The Harmonic Series DivergesProve the Harmonic Series diverges:
Compare the Series to the graph of .
…
So…
Since is decreasing, the Left Hand Riemann Sum is an over estimate.
Thus:
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The Harmonic Series DivergesProve the Harmonic Series diverges:
Compare the Series to the graph of .
…
So… We can find the value of the
improper integral:
Since diverges and , the Harmonic Series Diverges.
11
limb
xbdx
1lim ln
b
bx
lim ln ln1b
b
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The Harmonic Series Diverges Part 2
Justify that the Harmonic Series diverges another way:
Investigate the sum:121 1 1
3 4 1 1 1 15 6 7 8 1 1
9 16... ...12
12 1
2 12
By increasing the size of , we can make the sum of the infinite series as large was we desire.
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The Alternating Harmonic Series Converges
Justify the Alternating Harmonic Series converges:
1 12
13
14
15
0 12
1
Investigate and plot the sum:16
17 ...
The sum is bounded by 0.5 and 1.
Each Successive term in the sequence of partial
sums is between the two previous terms in
this sequence .
The sum must be between any two successive terms.
?S
We will find the actual value of the
sum soon.
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Arithmetic and Geometric SeriesAn Arithmetic Series has a constant difference between terms. (Similar to an Arithmetic Sequence.)Example:
A Geometric Series has a constant ratio between terms. (Similar to a Geometric Sequence.)Example:
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Arithmetic and Geometric SeriesBy the n-th Term Test, every Arithmetic Series diverges:
Some Geometric Series diverge and others converge:
Since Geometric Series occasionally converge, we will focus on them.
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Definition of a Geometric SeriesIn a geometric series each term is obtained from its preceding term by multiplying by the same number :
Examples: The previous examples are geometric.
∑𝑛=1
∞ 510𝑛
∑𝑛=1
∞
0.1(2)𝑛 ∑𝑛=1
∞
100 (0.5)𝑛
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White Board ChallengeFind the general term and the sum of the first 10 terms of the sequence:
18 1.5 nna
10 906.641s
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Finite Sum of a Geometric SeriesFind the sum of the first terms of a geometric series:
2 3 1... nS a ar ar ar ar 2 3 1... n nrS ar ar ar ar ar ________________________________
nS rS a ar 1 1 nS r a r
1
1
na rS
r
What happens to the sum as the value of n increases to infinity?
Multiply by r.
Subtract the two equations.
Solve for the sum.
Check with the
previous example.
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Infinite Sum of a Geometric Series
Consider :
1if r 1if r 1
1limna r
rnS
Diverges
1
1limna r
rnS
1 01lim arn
1ar
Depends on the value of r.
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Convergent Geometric Series
The geometric series converges if and only if . If the series converges, its sum is .
Example: Find the sum if it exists.1. 2. 3.
Where a is the first term and r is the constant ratio.
0.3a 0.3 11 .1 3S
0.030.3 0.1r
1125 2.5a 2.5
1 0.5 5S 12r
2a 12 /4/2 2r
Diverges