28 alternating series and conditional convergence
Transcript of 28 alternating series and conditional convergence
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Alternating Series and Conditional Convergence
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Alternating Series and Conditional Convergence
An alternating series is a series with alternating positive and negative terms.
![Page 3: 28 alternating series and conditional convergence](https://reader035.fdocuments.us/reader035/viewer/2022062709/558e93881a28ab52108b461c/html5/thumbnails/3.jpg)
Alternating Series and Conditional Convergence
An alternating series is a series with alternating positive and negative terms. Alternating series usally
are given as Σn=1
∞(-1)nan = -a1 + a2 – a3 + a4 …. or as Σ
n=1
∞(-1)n+1an = a1 – a2 + a3 – a4 … where an > 0
![Page 4: 28 alternating series and conditional convergence](https://reader035.fdocuments.us/reader035/viewer/2022062709/558e93881a28ab52108b461c/html5/thumbnails/4.jpg)
Alternating Series and Conditional Convergence
An alternating series is a series with alternating positive and negative terms. Alternating series usally
are given as Σn=1
∞(-1)nan = -a1 + a2 – a3 + a4 …. or as Σ
n=1
∞(-1)n+1an = a1 – a2 + a3 – a4 … where an > 0
From here on, we will write an alternating series as
Σn=1
∞an where the sequence {an} has alternating signs.
![Page 5: 28 alternating series and conditional convergence](https://reader035.fdocuments.us/reader035/viewer/2022062709/558e93881a28ab52108b461c/html5/thumbnails/5.jpg)
Alternating Series and Conditional Convergence
An alternating series is a series with alternating positive and negative terms. Alternating series usally
are given as Σn=1
∞(-1)nan = -a1 + a2 – a3 + a4 …. or as Σ
n=1
∞(-1)n+1an = a1 – a2 + a3 – a4 … where an > 0
Theorem (Alternating Series): Suppose {an} is a alternating sequence and lim an = 0, then the
alternating series converges. n∞
From here on, we will write an alternating series as
Σn=1
∞an where the sequence {an} has alternating signs.
Σn=1
∞an
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Alternating Series and Conditional Convergence
An alternating series is a series with alternating positive and negative terms. Alternating series usally
are given as Σn=1
∞(-1)nan = -a1 + a2 – a3 + a4 …. or as Σ
n=1
∞(-1)n+1an = a1 – a2 + a3 – a4 … where an > 0
Theorem (Alternating Series): Suppose {an} is a alternating sequence and lim an = 0, then the
alternating series converges. n∞
We will use the alternating harmonic series
to demonstrate the argument for the theorem.
12
13 ...1 +Σ
n=1(-1)n+1 = 1
n∞ 1
4 – –
From here on, we will write an alternating series as
Σn=1
∞an where the sequence {an} has alternating signs.
Σn=1
∞an
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Alternating Series and Conditional Convergence
12
13 ...1 +Σ
n=1(-1)n+1 = 1
n∞ 1
4 – –The convergence of
may be demonstrated on the real line.
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Alternating Series and Conditional Convergence
12
13 ...1 +Σ
n=1(-1)n+1 = 1
n∞ 1
4 – –The convergence of
may be demonstrated on the real line.
Let Sn = be the partial sum. 12
13 ...1 + 1
4 – – 1
n±
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Alternating Series and Conditional Convergence
12
13 ...1 +Σ
n=1(-1)n+1 = 1
n∞ 1
4 – –The convergence of
may be demonstrated on the real line.
Let Sn = be the partial sum. 12
13 ...1 + 1
4 – – 1
n±
We may visualize Sn geometrically as following:
0 1
S11
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Alternating Series and Conditional Convergence
12
13 ...1 +Σ
n=1(-1)n+1 = 1
n∞ 1
4 – –The convergence of
may be demonstrated on the real line.
Let Sn = be the partial sum. 12
13 ...1 + 1
4 – – 1
n±
We may visualize Sn geometrically as following:
0 1
S11
S21/2
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Alternating Series and Conditional Convergence
12
13 ...1 +Σ
n=1(-1)n+1 = 1
n∞ 1
4 – –The convergence of
may be demonstrated on the real line.
Let Sn = be the partial sum. 12
13 ...1 + 1
4 – – 1
n±
We may visualize Sn geometrically as following:
0 1
S11
S21/2
S31/3
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Alternating Series and Conditional Convergence
12
13 ...1 +Σ
n=1(-1)n+1 = 1
n∞ 1
4 – –The convergence of
may be demonstrated on the real line.
Let Sn = be the partial sum. 12
13 ...1 + 1
4 – – 1
n±
We may visualize Sn geometrically as following:
0 1
S11
S21/2
S31/3
S41/4
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Alternating Series and Conditional Convergence
12
13 ...1 +Σ
n=1(-1)n+1 = 1
n∞ 1
4 – –The convergence of
may be demonstrated on the real line.
Let Sn = be the partial sum. 12
13 ...1 + 1
4 – – 1
n±
We may visualize Sn geometrically as following:
0 1
S11
S21/2
S31/3
S41/4
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Alternating Series and Conditional Convergence
12
13 ...1 +Σ
n=1(-1)n+1 = 1
n∞ 1
4 – –The convergence of
may be demonstrated on the real line.
Let Sn = be the partial sum. 12
13 ...1 + 1
4 – – 1
n±
We may visualize Sn geometrically as following:
0 1
S11
S21/2
S31/3
S41/4
Since 0, the red dot must approach a limit
which is
1n
Σn=1
(-1)n+1 . 1n
∞
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Alternating Series and Conditional Convergence
12
13 ...1 +Σ
n=1(-1)n+1 = 1
n∞ 1
4 – –The convergence of
may be demonstrated on the real line.
Let Sn = be the partial sum. 12
13 ...1 + 1
4 – – 1
n±
We may visualize Sn geometrically as following:
0 1
S11
S21/2
S31/3
S41/4
Since 0, the red dot must approach a limit
which is
1n
Σn=1
(-1)n+1 . 1n
∞(Later we'll see its Ln(2) 0.59)
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Alternating Series and Conditional Convergence
Σn=1
∞Thus given an alternating sequence an 0,
an converges.
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Alternating Series and Conditional Convergence
Σn=1
∞Thus given an alternating sequence an 0,
an converges. However, to be precise, we define
two notions of convergence for alternating series:
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Alternating Series and Conditional Convergence
Σn=1
∞Thus given an alternating sequence an 0,
an
If also converges, we say the alternating series
converges absolutely.
Σn=1
|an|∞
converges. However, to be precise, we define
two notions of convergence for alternating series:
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Alternating Series and Conditional Convergence
Σn=1
∞Thus given an alternating sequence an 0,
an
If also converges, we say the alternating series
converges absolutely.
Σn=1
|an|∞
converges. However, to be precise, we define
If also diverges, we say the alternating series
converges conditionally.
Σn=1
|an|∞
two notions of convergence for alternating series:
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Alternating Series and Conditional Convergence
Σn=1
∞Thus given an alternating sequence an 0,
an
If also converges, we say the alternating series
converges absolutely.
Σn=1
|an|∞
converges. However, to be precise, we define
If also diverges, we say the alternating series
converges conditionally.
Σn=1
|an|
The alternating geometric series , | r | < 1 1rn(-1)nΣ
n=1
∞
∞
two notions of convergence for alternating series:
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Alternating Series and Conditional Convergence
Σn=1
∞Thus given an alternating sequence an 0,
an
If also converges, we say the alternating series
converges absolutely.
Σn=1
|an|∞
converges. However, to be precise, we define
If also diverges, we say the alternating series
converges conditionally.
Σn=1
|an|
The alternating geometric series , | r | < 1
and the alternating p-series where p > 1
converges absolutely
1np(-1)n
1rn(-1)nΣ
n=1
∞
Σn=1
∞
∞
two notions of convergence for alternating series:
![Page 22: 28 alternating series and conditional convergence](https://reader035.fdocuments.us/reader035/viewer/2022062709/558e93881a28ab52108b461c/html5/thumbnails/22.jpg)
Alternating Series and Conditional Convergence
Σn=1
∞Thus given an alternating sequence an 0,
an
If also converges, we say the alternating series
converges absolutely.
Σn=1
|an|∞
converges. However, to be precise, we define
If also diverges, we say the alternating series
converges conditionally.
Σn=1
|an|
The alternating geometric series , | r | < 1
and the alternating p-series where p > 1
converges absolutely (because converges).
1np(-1)n
1rn(-1)nΣ
n=1
∞
Σn=1
∞
∞
two notions of convergence for alternating series:
1rn,Σ
n=1
∞ 1npΣ
n=1
∞
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Alternating Series and Conditional Convergence
The alternating harmonic series converges1n(-1)nΣ
n=1
∞
conditionally (because diverges).1nΣ
n=1
∞
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Alternating Series and Conditional Convergence
The alternating harmonic series converges1n(-1)nΣ
n=1
∞
conditionally (because diverges).1nΣ
n=1
∞
The alternating p-series , p < 1 converges1np(-1)nΣ
n=1
∞
conditionally (because diverges).1npΣ
∞
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Alternating Series and Conditional Convergence
The alternating harmonic series converges1n(-1)nΣ
n=1
∞
conditionally (because diverges).1nΣ
n=1
∞
The alternating p-series , p < 1 converges1np(-1)nΣ
n=1
∞
conditionally (because diverges).1npΣ
n=1
∞
Hence when we answer questions about the convergence of alternaing series, we always identify if it converges absolutely or condiditonally
![Page 26: 28 alternating series and conditional convergence](https://reader035.fdocuments.us/reader035/viewer/2022062709/558e93881a28ab52108b461c/html5/thumbnails/26.jpg)
Alternating Series and Conditional Convergence
The alternating harmonic series converges1n(-1)nΣ
n=1
∞
conditionally (because diverges).1nΣ
n=1
∞
The alternating p-series , p < 1 converges1np(-1)nΣ
n=1
∞
conditionally (because diverges).1npΣ
n=1
∞
Hence when we answer questions about the convergence of alternaing series, we always identify if it converges absolutely or condiditonally(or if it diverges, i.e an doesn't go to 0).
![Page 27: 28 alternating series and conditional convergence](https://reader035.fdocuments.us/reader035/viewer/2022062709/558e93881a28ab52108b461c/html5/thumbnails/27.jpg)
Alternating Series and Conditional Convergence
The alternating harmonic series converges1n(-1)nΣ
n=1
∞
conditionally (because diverges).1nΣ
n=1
∞
The alternating p-series , p < 1 converges1np(-1)nΣ
n=1
∞
conditionally (because diverges).1npΣ
n=1
∞
Hence when we answer questions about the convergence of alternaing series, we always identify if it converges absolutely or condiditonally(or if it diverges, i.e an doesn't go to 0).
Example: Discuss the convergence of the following theorem. a. (-1)nΣ
n=1
∞
= -1 + 1 – 1 + 1 …
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Alternating Series and Conditional Convergence
The alternating harmonic series converges1n(-1)nΣ
n=1
∞
conditionally (because diverges).1nΣ
n=1
∞
The alternating p-series , p < 1 converges1np(-1)nΣ
n=1
∞
conditionally (because diverges).1npΣ
n=1
∞
Hence when we answer questions about the convergence of alternaing series, we always identify if it converges absolutely or condiditonally(or if it diverges, i.e an doesn't go to 0).
Example: Discuss the convergence of the following theorem. a. (-1)nΣ
n=1
∞
= -1 + 1 – 1 + 1 … diverges since an does not
converges to 0.
![Page 29: 28 alternating series and conditional convergence](https://reader035.fdocuments.us/reader035/viewer/2022062709/558e93881a28ab52108b461c/html5/thumbnails/29.jpg)
Example:
b. an = 3n n2 + n – 1.
Alternating Series and Conditional Convergence
(-1)n
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Example:
b. an = 3n n2 + n – 1.
Alternating Series and Conditional Convergence
(-1)n Since an 0, converges.Σn=1
∞an
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Example:
b. an = 3n n2 + n – 1.
Alternating Series and Conditional Convergence
(-1)n
We need to identify what type of convergence it is.
Since an 0, converges.Σn=1
∞an
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Example:
b. an = 3n n2 + n – 1.
Alternating Series and Conditional Convergence
(-1)n
We need to identify what type of convergence it is.
The dominating terms are 3n n2 .
Since an 0, converges.Σn=1
∞an
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Example:
b. an = 3n n2 + n – 1.
Alternating Series and Conditional Convergence
(-1)n
We need to identify what type of convergence it is.
The dominate terms are 3n n2 .
Compare this to an.
Since an 0, converges.Σn=1
∞an
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Example:
b. an = 3n n2 + n – 1.
Alternating Series and Conditional Convergence
(-1)n
We need to identify what type of convergence it is.
The dominate terms are 3n n2 .
Compare this to an.
n∞Lim 3n
n2 + n – 1 3n n2
Since an 0, converges.Σn=1
∞an
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Example:
b. an = 3n n2 + n – 1.
Alternating Series and Conditional Convergence
(-1)n
We need to identify what type of convergence it is.
The dominate terms are 3n n2 .
Compare this to an.
n∞Lim 3n
n2 + n – 1 3n n2
= 1
Since an 0, converges.Σn=1
∞an
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Example:
b. an = 3n n2 + n – 1.
So {an} and are almost-multiple of each other.
Alternating Series and Conditional Convergence
(-1)n
We need to identify what type of convergence it is.
The dominate terms are 3n n2 .
Compare this to an.
n∞Lim 3n
n2 + n – 1 3n n2
= 1
3n n2
Since an 0, converges.Σn=1
∞an
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Example:
b. an = 3n n2 + n – 1.
So {an} and are almost-multiple of each other.
Alternating Series and Conditional Convergence
(-1)n
We need to identify what type of convergence it is.
The dominate terms are 3n n2 .
Compare this to an.
n∞Lim 3n
n2 + n – 1 3n n2
= 1
3n n2
But Σn=1
∞ 3n n2 = Σ
n=1
∞ 3 n diverges.
Since an 0, converges.Σn=1
∞an
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Example:
b. an = 3n n2 + n – 1.
So {an} and are almost-multiple of each other.
Alternating Series and Conditional Convergence
(-1)n
We need to identify what type of convergence it is.
The dominate terms are 3n n2 .
Compare this to an.
n∞Lim 3n
n2 + n – 1 3n n2
= 1
3n n2
But Σn=1
∞ 3n n2 = Σ
n=1
∞ 3 n diverges.
Therefore Σn=1
3n n2 + n – 1
diverges.
Since an 0, converges.Σn=1
∞an
∞
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Example:
b. an = 3n n2 + n – 1.
So {an} and are almost-multiple of each other.
Alternating Series and Conditional Convergence
(-1)n
We need to identify what type of convergence it is.
The dominate terms are 3n n2 .
Compare this to an.
n∞Lim 3n
n2 + n – 1 3n n2
= 1
3n n2
But Σn=1
∞ 3n n2 = Σ
n=1
∞ 3 n diverges.
Therefore Σn=1
3n n2 + n – 1
diverges.
So
Since an 0, converges.Σn=1
∞
an
Σn=1
3n n2 + n – 1
(-1)n
∞
∞converges conditionally.
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Alternating Series and Conditional Convergence
Summary: Given an alternating sequence {an}
![Page 41: 28 alternating series and conditional convergence](https://reader035.fdocuments.us/reader035/viewer/2022062709/558e93881a28ab52108b461c/html5/thumbnails/41.jpg)
Alternating Series and Conditional Convergence
Summary: Given an alternating sequence {an}1. check if an 0, if no, the series diverges
![Page 42: 28 alternating series and conditional convergence](https://reader035.fdocuments.us/reader035/viewer/2022062709/558e93881a28ab52108b461c/html5/thumbnails/42.jpg)
Alternating Series and Conditional Convergence
Summary: Given an alternating sequence {an}1. check if an 0, if no, the series diverges2. if yes, check what type of convergence by
investigating the convergence of
Σn=1
∞|an|.
![Page 43: 28 alternating series and conditional convergence](https://reader035.fdocuments.us/reader035/viewer/2022062709/558e93881a28ab52108b461c/html5/thumbnails/43.jpg)
Alternating Series and Conditional Convergence
Summary: Given an alternating sequence {an}1. check if an 0, if no, the series diverges2. if yes, check what type of convergence by
investigating the convergence of
If it converges, the alternating series converges absolutely.
Σn=1
∞|an|.
![Page 44: 28 alternating series and conditional convergence](https://reader035.fdocuments.us/reader035/viewer/2022062709/558e93881a28ab52108b461c/html5/thumbnails/44.jpg)
Alternating Series and Conditional Convergence
Summary: Given an alternating sequence {an}1. check if an 0, if no, the series diverges2. if yes, check what type of convergence by
investigating the convergence of
If it converges, the alternating series converges absolutely. If it diverges, the alternating series converges conditionally.
Σn=1
∞|an|.