25 the harmonic series and the integral test
TRANSCRIPT
![Page 1: 25 the harmonic series and the integral test](https://reader035.vdocument.in/reader035/viewer/2022062514/558e94cb1a28ab3b108b4722/html5/thumbnails/1.jpg)
The Harmonic Series and the Integral Test
![Page 2: 25 the harmonic series and the integral test](https://reader035.vdocument.in/reader035/viewer/2022062514/558e94cb1a28ab3b108b4722/html5/thumbnails/2.jpg)
The Harmonic Series and the Integral Test
If we add infinitely many terms and obtain a finite sum, it must be the case that the terms get smaller and smaller and goes to zero.
![Page 3: 25 the harmonic series and the integral test](https://reader035.vdocument.in/reader035/viewer/2022062514/558e94cb1a28ab3b108b4722/html5/thumbnails/3.jpg)
Theorem:
The Harmonic Series and the Integral Test
If we add infinitely many terms and obtain a finite sum, it must be the case that the terms get smaller and smaller and goes to zero.
If = a1 + a2 + a3 + … = L is a Σi=1
∞ai
convergent series, then lim an = 0.n∞
![Page 4: 25 the harmonic series and the integral test](https://reader035.vdocument.in/reader035/viewer/2022062514/558e94cb1a28ab3b108b4722/html5/thumbnails/4.jpg)
Proof: Let = a1 + a2 .. = L be a convergent series. Σi=1
∞ai
Theorem:
The Harmonic Series and the Integral Test
If we add infinitely many terms and obtain a finite sum, it must be the case that the terms get smaller and smaller and goes to zero.
If = a1 + a2 + a3 + … = L is a Σi=1
∞ai
convergent series, then lim an = 0.n∞
![Page 5: 25 the harmonic series and the integral test](https://reader035.vdocument.in/reader035/viewer/2022062514/558e94cb1a28ab3b108b4722/html5/thumbnails/5.jpg)
Proof: Let = a1 + a2 .. = L be a convergent series. Σi=1
∞ai
Theorem:
This means the for the sequence of partial sums,
lim sn = lim (a1 + a2 + … + an) = L converges.
The Harmonic Series and the Integral Test
If we add infinitely many terms and obtain a finite sum, it must be the case that the terms get smaller and smaller and goes to zero.
n∞
If = a1 + a2 + a3 + … = L is a Σi=1
∞ai
convergent series, then lim an = 0.n∞
![Page 6: 25 the harmonic series and the integral test](https://reader035.vdocument.in/reader035/viewer/2022062514/558e94cb1a28ab3b108b4722/html5/thumbnails/6.jpg)
Proof: Let = a1 + a2 .. = L be a convergent series. Σi=1
∞ai
Theorem:
The Harmonic Series and the Integral Test
If we add infinitely many terms and obtain a finite sum, it must be the case that the terms get smaller and smaller and goes to zero.
n∞
lim sn-1 = lim (a1 + a2 +..+ an-1) n∞ n∞
If = a1 + a2 + a3 + … = L is a Σi=1
∞ai
convergent series, then lim an = 0.n∞
On the other hand,
This means the for the sequence of partial sums,
lim sn = lim (a1 + a2 + … + an) = L converges.
![Page 7: 25 the harmonic series and the integral test](https://reader035.vdocument.in/reader035/viewer/2022062514/558e94cb1a28ab3b108b4722/html5/thumbnails/7.jpg)
Proof: Let = a1 + a2 .. = L be a convergent series. Σi=1
∞ai
Theorem:
The Harmonic Series and the Integral Test
If we add infinitely many terms and obtain a finite sum, it must be the case that the terms get smaller and smaller and goes to zero.
n∞
lim sn-1 = lim (a1 + a2 +..+ an-1) = L.n∞ n∞
If = a1 + a2 + a3 + … = L is a Σi=1
∞ai
convergent series, then lim an = 0.n∞
On the other hand,
This means the for the sequence of partial sums,
lim sn = lim (a1 + a2 + … + an) = L converges.
![Page 8: 25 the harmonic series and the integral test](https://reader035.vdocument.in/reader035/viewer/2022062514/558e94cb1a28ab3b108b4722/html5/thumbnails/8.jpg)
Proof: Let = a1 + a2 .. = L be a convergent series. Σi=1
∞ai
Theorem:
The Harmonic Series and the Integral Test
If we add infinitely many terms and obtain a finite sum, it must be the case that the terms get smaller and smaller and goes to zero.
n∞
lim sn-1 = lim (a1 + a2 +..+ an-1) = L.n∞ n∞
Hence lim sn – sn-1 n∞
If = a1 + a2 + a3 + … = L is a Σi=1
∞ai
convergent series, then lim an = 0.n∞
On the other hand,
This means the for the sequence of partial sums,
lim sn = lim (a1 + a2 + … + an) = L converges.
![Page 9: 25 the harmonic series and the integral test](https://reader035.vdocument.in/reader035/viewer/2022062514/558e94cb1a28ab3b108b4722/html5/thumbnails/9.jpg)
Proof: Let = a1 + a2 .. = L be a convergent series. Σi=1
∞ai
Theorem:
The Harmonic Series and the Integral Test
If we add infinitely many terms and obtain a finite sum, it must be the case that the terms get smaller and smaller and goes to zero.
n∞
lim sn-1 = lim (a1 + a2 +..+ an-1) = L.n∞ n∞
Hence lim sn – sn-1 = lim an n∞ n∞
If = a1 + a2 + a3 + … = L is a Σi=1
∞ai
convergent series, then lim an = 0.n∞
On the other hand,
This means the for the sequence of partial sums,
lim sn = lim (a1 + a2 + … + an) = L converges.
![Page 10: 25 the harmonic series and the integral test](https://reader035.vdocument.in/reader035/viewer/2022062514/558e94cb1a28ab3b108b4722/html5/thumbnails/10.jpg)
Proof: Let = a1 + a2 .. = L be a convergent series. Σi=1
∞ai
Theorem:
The Harmonic Series and the Integral Test
If we add infinitely many terms and obtain a finite sum, it must be the case that the terms get smaller and smaller and goes to zero.
n∞
lim sn-1 = lim (a1 + a2 +..+ an-1) = L.n∞ n∞
Hence lim sn – sn-1 = lim an = L – L = 0.n∞ n∞
If = a1 + a2 + a3 + … = L is a Σi=1
∞ai
convergent series, then lim an = 0.n∞
On the other hand,
This means the for the sequence of partial sums,
lim sn = lim (a1 + a2 + … + an) = L converges.
![Page 11: 25 the harmonic series and the integral test](https://reader035.vdocument.in/reader035/viewer/2022062514/558e94cb1a28ab3b108b4722/html5/thumbnails/11.jpg)
The Harmonic Series and the Integral Test
However the fact that lim an 0 does not guarantee that their sum CGs to a finite number.
![Page 12: 25 the harmonic series and the integral test](https://reader035.vdocument.in/reader035/viewer/2022062514/558e94cb1a28ab3b108b4722/html5/thumbnails/12.jpg)
Example: The sequence 1,
The Harmonic Series and the Integral Test
However the fact that lim an 0 does not guarantee that their sum CGs to a finite number.
12 ,
12 ,
13 ,
13 ,
13 ,
14 ,
14 ,
14 ,
14 , 0, 1
5 , ..
![Page 13: 25 the harmonic series and the integral test](https://reader035.vdocument.in/reader035/viewer/2022062514/558e94cb1a28ab3b108b4722/html5/thumbnails/13.jpg)
Example: The sequence 1,
The Harmonic Series and the Integral Test
However the fact that lim an 0 does not guarantee that their sum CGs to a finite number.
12 ,
12 ,
13 ,
13 ,
13 ,
14 ,
14 ,
14 ,
14 , 0, 1
5 , ..
but their sum
![Page 14: 25 the harmonic series and the integral test](https://reader035.vdocument.in/reader035/viewer/2022062514/558e94cb1a28ab3b108b4722/html5/thumbnails/14.jpg)
Example: The sequence 1,
The Harmonic Series and the Integral Test
However the fact that lim an 0 does not guarantee that their sum CGs to a finite number.
12 ,
12 ,
13 ,
13 ,
13 ,
14 ,
14 ,
14 ,
14 , 0, 1
5 , ..
but their sum 1
![Page 15: 25 the harmonic series and the integral test](https://reader035.vdocument.in/reader035/viewer/2022062514/558e94cb1a28ab3b108b4722/html5/thumbnails/15.jpg)
Example: The sequence 1,
The Harmonic Series and the Integral Test
However the fact that lim an 0 does not guarantee that their sum CGs to a finite number.
12 ,
12 ,
13 ,
13 ,
13 ,
14 ,
14 ,
14 ,
14 , 0, 1
5 , ..
but their sum 1+ 12 + 12
![Page 16: 25 the harmonic series and the integral test](https://reader035.vdocument.in/reader035/viewer/2022062514/558e94cb1a28ab3b108b4722/html5/thumbnails/16.jpg)
Example: The sequence 1,
The Harmonic Series and the Integral Test
However the fact that lim an 0 does not guarantee that their sum CGs to a finite number.
12 ,
12 ,
13 ,
13 ,
13 ,
14 ,
14 ,
14 ,
14 , 0, 1
5 , ..
but their sum 1+ 12 + 12
13
13
13+ + +
![Page 17: 25 the harmonic series and the integral test](https://reader035.vdocument.in/reader035/viewer/2022062514/558e94cb1a28ab3b108b4722/html5/thumbnails/17.jpg)
Example: The sequence 1,
The Harmonic Series and the Integral Test
However the fact that lim an 0 does not guarantee that their sum CGs to a finite number.
12 ,
12 ,
13 ,
13 ,
13 ,
14 ,
14 ,
14 ,
14 , 0, 1
5 , ..
but their sum 1+ 12 + 12
13
13
13
14
14
14
14+ + + + + + +
![Page 18: 25 the harmonic series and the integral test](https://reader035.vdocument.in/reader035/viewer/2022062514/558e94cb1a28ab3b108b4722/html5/thumbnails/18.jpg)
Example: The sequence 1,
The Harmonic Series and the Integral Test
However the fact that lim an 0 does not guarantee that their sum CGs to a finite number.
12 ,
12 ,
13 ,
13 ,
13 ,
14 ,
14 ,
14 ,
14 , 0, 1
5 , ..
but their sum 1+ 12 + 12
13
13
13
14
14
14
14
15 ..+ + + + + + + + = ∞
![Page 19: 25 the harmonic series and the integral test](https://reader035.vdocument.in/reader035/viewer/2022062514/558e94cb1a28ab3b108b4722/html5/thumbnails/19.jpg)
Example: The sequence 1,
The Harmonic Series and the Integral Test
However the fact that lim an 0 does not guarantee that their sum CGs to a finite number.
12 ,
12 ,
13 ,
13 ,
13 ,
14 ,
14 ,
14 ,
14 , 0, 1
5 , ..
but their sum 1+ 12 + 12
13
13
13
14
14
14
14
15 ..+ + + + + + + + = ∞
An important sequence that goes to 0 but sums to ∞is the harmonic sequence: {1/n} = 1
2 ,13 ,
14 , ..1,{ }
![Page 20: 25 the harmonic series and the integral test](https://reader035.vdocument.in/reader035/viewer/2022062514/558e94cb1a28ab3b108b4722/html5/thumbnails/20.jpg)
Example: The sequence 1,
The Harmonic Series and the Integral Test
However the fact that lim an 0 does not guarantee that their sum CGs to a finite number.
12 ,
12 ,
13 ,
13 ,
13 ,
14 ,
14 ,
14 ,
14 , 0, 1
5 , ..
but their sum 1+ 12 + 12
13
13
13
14
14
14
14
15 ..+ + + + + + + + = ∞
An important sequence that goes to 0 but sums to ∞is the harmonic sequence: {1/n} = 1
2 ,13 ,
14 , ..1,{ }
To see that they sum to ∞, sum in blocks as shown:
![Page 21: 25 the harmonic series and the integral test](https://reader035.vdocument.in/reader035/viewer/2022062514/558e94cb1a28ab3b108b4722/html5/thumbnails/21.jpg)
Example: The sequence 1,
The Harmonic Series and the Integral Test
However the fact that lim an 0 does not guarantee that their sum CGs to a finite number.
12 ,
12 ,
13 ,
13 ,
13 ,
14 ,
14 ,
14 ,
14 , 0, 1
5 , ..
but their sum 1+ 12 + 12
13
13
13
14
14
14
14
15 ..+ + + + + + + + = ∞
An important sequence that goes to 0 but sums to ∞is the harmonic sequence: {1/n} = 1
2 ,13 ,
14 , ..1,{ }
To see that they sum to ∞, sum in blocks as shown:
1
![Page 22: 25 the harmonic series and the integral test](https://reader035.vdocument.in/reader035/viewer/2022062514/558e94cb1a28ab3b108b4722/html5/thumbnails/22.jpg)
Example: The sequence 1,
The Harmonic Series and the Integral Test
However the fact that lim an 0 does not guarantee that their sum CGs to a finite number.
12 ,
12 ,
13 ,
13 ,
13 ,
14 ,
14 ,
14 ,
14 , 0, 1
5 , ..
but their sum 1+ 12 + 12
13
13
13
14
14
14
14
15 ..+ + + + + + + + = ∞
An important sequence that goes to 0 but sums to ∞is the harmonic sequence: {1/n} = 1
2 ,13 ,
14 , ..1,{ }
To see that they sum to ∞, sum in blocks as shown:12
13
110...1 + + + +
![Page 23: 25 the harmonic series and the integral test](https://reader035.vdocument.in/reader035/viewer/2022062514/558e94cb1a28ab3b108b4722/html5/thumbnails/23.jpg)
Example: The sequence 1,
The Harmonic Series and the Integral Test
However the fact that lim an 0 does not guarantee that their sum CGs to a finite number.
12 ,
12 ,
13 ,
13 ,
13 ,
14 ,
14 ,
14 ,
14 , 0, 1
5 , ..
but their sum 1+ 12 + 12
13
13
13
14
14
14
14
15 ..+ + + + + + + + = ∞
An important sequence that goes to 0 but sums to ∞is the harmonic sequence: {1/n} = 1
2 ,13 ,
14 , ..1,{ }
To see that they sum to ∞, sum in blocks as shown:12
13
110...1 + + + +
> 910
![Page 24: 25 the harmonic series and the integral test](https://reader035.vdocument.in/reader035/viewer/2022062514/558e94cb1a28ab3b108b4722/html5/thumbnails/24.jpg)
Example: The sequence 1,
The Harmonic Series and the Integral Test
However the fact that lim an 0 does not guarantee that their sum CGs to a finite number.
12 ,
12 ,
13 ,
13 ,
13 ,
14 ,
14 ,
14 ,
14 , 0, 1
5 , ..
but their sum 1+ 12 + 12
13
13
13
14
14
14
14
15 ..+ + + + + + + + = ∞
An important sequence that goes to 0 but sums to ∞is the harmonic sequence: {1/n} = 1
2 ,13 ,
14 , ..1,{ }
To see that they sum to ∞, sum in blocks as shown:12
13
110...1 + + + + 1
11+ 1100... +
> 910
![Page 25: 25 the harmonic series and the integral test](https://reader035.vdocument.in/reader035/viewer/2022062514/558e94cb1a28ab3b108b4722/html5/thumbnails/25.jpg)
Example: The sequence 1,
The Harmonic Series and the Integral Test
However the fact that lim an 0 does not guarantee that their sum CGs to a finite number.
12 ,
12 ,
13 ,
13 ,
13 ,
14 ,
14 ,
14 ,
14 , 0, 1
5 , ..
but their sum 1+ 12 + 12
13
13
13
14
14
14
14
15 ..+ + + + + + + + = ∞
An important sequence that goes to 0 but sums to ∞is the harmonic sequence: {1/n} = 1
2 ,13 ,
14 , ..1,{ }
To see that they sum to ∞, sum in blocks as shown:12
13
110...1 + + + + 1
11+ 1100... +
> 910 > 90
100
![Page 26: 25 the harmonic series and the integral test](https://reader035.vdocument.in/reader035/viewer/2022062514/558e94cb1a28ab3b108b4722/html5/thumbnails/26.jpg)
Example: The sequence 1,
The Harmonic Series and the Integral Test
However the fact that lim an 0 does not guarantee that their sum CGs to a finite number.
12 ,
12 ,
13 ,
13 ,
13 ,
14 ,
14 ,
14 ,
14 , 0, 1
5 , ..
but their sum 1+ 12 + 12
13
13
13
14
14
14
14
15 ..+ + + + + + + + = ∞
An important sequence that goes to 0 but sums to ∞is the harmonic sequence: {1/n} = 1
2 ,13 ,
14 , ..1,{ }
To see that they sum to ∞, sum in blocks as shown:12
13
110...1 + + + + 1
11+ 1100... +
> 910 > 90
100= 910
![Page 27: 25 the harmonic series and the integral test](https://reader035.vdocument.in/reader035/viewer/2022062514/558e94cb1a28ab3b108b4722/html5/thumbnails/27.jpg)
Example: The sequence 1,
The Harmonic Series and the Integral Test
However the fact that lim an 0 does not guarantee that their sum CGs to a finite number.
12 ,
12 ,
13 ,
13 ,
13 ,
14 ,
14 ,
14 ,
14 , 0, 1
5 , ..
but their sum 1+ 12 + 12
13
13
13
14
14
14
14
15 ..+ + + + + + + + = ∞
An important sequence that goes to 0 but sums to ∞is the harmonic sequence: {1/n} = 1
2 ,13 ,
14 , ..1,{ }
To see that they sum to ∞, sum in blocks as shown:12
13
110...1 + + + + 1
11+ 1100... + 1
101+ 11000... +
> 910 > 90
100= 910
![Page 28: 25 the harmonic series and the integral test](https://reader035.vdocument.in/reader035/viewer/2022062514/558e94cb1a28ab3b108b4722/html5/thumbnails/28.jpg)
Example: The sequence 1,
The Harmonic Series and the Integral Test
However the fact that lim an 0 does not guarantee that their sum CGs to a finite number.
12 ,
12 ,
13 ,
13 ,
13 ,
14 ,
14 ,
14 ,
14 , 0, 1
5 , ..
but their sum 1+ 12 + 12
13
13
13
14
14
14
14
15 ..+ + + + + + + + = ∞
An important sequence that goes to 0 but sums to ∞is the harmonic sequence: {1/n} = 1
2 ,13 ,
14 , ..1,{ }
To see that they sum to ∞, sum in blocks as shown:12
13
110...1 + + + + 1
11+ 1100... + 1
101+ 11000... +
> 910 > 90
100= 910 > 1000
900
![Page 29: 25 the harmonic series and the integral test](https://reader035.vdocument.in/reader035/viewer/2022062514/558e94cb1a28ab3b108b4722/html5/thumbnails/29.jpg)
Example: The sequence 1,
The Harmonic Series and the Integral Test
However the fact that lim an 0 does not guarantee that their sum CGs to a finite number.
12 ,
12 ,
13 ,
13 ,
13 ,
14 ,
14 ,
14 ,
14 , 0, 1
5 , ..
but their sum 1+ 12 + 12
13
13
13
14
14
14
14
15 ..+ + + + + + + + = ∞
An important sequence that goes to 0 but sums to ∞is the harmonic sequence: {1/n} = 1
2 ,13 ,
14 , ..1,{ }
To see that they sum to ∞, sum in blocks as shown:12
13
110...1 + + + + 1
11+ 1100... + 1
101+ 11000... +
> 910 > 90
100= 910 > 1000 = 10
900 9
![Page 30: 25 the harmonic series and the integral test](https://reader035.vdocument.in/reader035/viewer/2022062514/558e94cb1a28ab3b108b4722/html5/thumbnails/30.jpg)
Example: The sequence 1,
The Harmonic Series and the Integral Test
However the fact that lim an 0 does not guarantee that their sum CGs to a finite number.
12 ,
12 ,
13 ,
13 ,
13 ,
14 ,
14 ,
14 ,
14 , 0, 1
5 , ..
but their sum 1+ 12 + 12
13
13
13
14
14
14
14
15 ..+ + + + + + + + = ∞
An important sequence that goes to 0 but sums to ∞is the harmonic sequence: {1/n} = 1
2 ,13 ,
14 , ..1,{ }
To see that they sum to ∞, sum in blocks as shown:12
13
110...1 + + + + 1
11+ 1100... + 1
101+ 11000... + + …
> 910 > 90
100= 910 > 1000 = 10
900 9
= ∞
![Page 31: 25 the harmonic series and the integral test](https://reader035.vdocument.in/reader035/viewer/2022062514/558e94cb1a28ab3b108b4722/html5/thumbnails/31.jpg)
Example: The sequence 1,
The Harmonic Series and the Integral Test
However the fact that lim an 0 does not guarantee that their sum CGs to a finite number.
12 ,
12 ,
13 ,
13 ,
13 ,
14 ,
14 ,
14 ,
14 , 0, 1
5 , ..
but their sum 1+ 12 + 12
13
13
13
14
14
14
14
15 ..+ + + + + + + + = ∞
An important sequence that goes to 0 but sums to ∞is the harmonic sequence: {1/n} = 1
2 ,13 ,
14 , ..1,{ }
To see that they sum to ∞, sum in blocks as shown:12
13
110...1 + + + + 1
11+ 1100... + 1
101+ 11000... + + …
> 910 > 90
100= 910 > 1000 = 10
900 9
= ∞
Hence the harmonic series DGs.
![Page 32: 25 the harmonic series and the integral test](https://reader035.vdocument.in/reader035/viewer/2022062514/558e94cb1a28ab3b108b4722/html5/thumbnails/32.jpg)
The Harmonic Series and the Integral Test
The following theorem and theorems in the next section give various methods of determining if a series is convergent or divergent.
![Page 33: 25 the harmonic series and the integral test](https://reader035.vdocument.in/reader035/viewer/2022062514/558e94cb1a28ab3b108b4722/html5/thumbnails/33.jpg)
The Harmonic Series and the Integral Test
The following theorem and theorems in the next section give various methods of determining if a series is convergent or divergent.
We shall assume all series are positive series, i.e. all terms in the series are positive unless stated otherwise.
![Page 34: 25 the harmonic series and the integral test](https://reader035.vdocument.in/reader035/viewer/2022062514/558e94cb1a28ab3b108b4722/html5/thumbnails/34.jpg)
Σi=1
∞
ai
Theorem:
The Harmonic Series and the Integral Test
The following theorem and theorems in the next section give various methods of determining if a series is convergent or divergent.
(Integral Test) If an = f(n) > 0, then
CGs if and only if
We shall assume all series are positive series, i.e. all terms in the series are positive unless stated otherwise.
∫1 f(x) dx CGs. ∞
![Page 35: 25 the harmonic series and the integral test](https://reader035.vdocument.in/reader035/viewer/2022062514/558e94cb1a28ab3b108b4722/html5/thumbnails/35.jpg)
Σi=1
∞
ai
Theorem:
The Harmonic Series and the Integral Test
The following theorem and theorems in the next section give various methods of determining if a series is convergent or divergent.
(Integral Test) If an = f(n) > 0, then
CGs if and only if
We shall assume all series are positive series, i.e. all terms in the series are positive unless stated otherwise.
∫1 f(x) dx CGs. ∞
Combine this with the p-theorem from before, we have the following theorem about the convergence of the p-series:
![Page 36: 25 the harmonic series and the integral test](https://reader035.vdocument.in/reader035/viewer/2022062514/558e94cb1a28ab3b108b4722/html5/thumbnails/36.jpg)
Σi=1
Theorem:
The Harmonic Series and the Integral Test
(p-series)
CGs if and only if p > 1. ∞
np1
![Page 37: 25 the harmonic series and the integral test](https://reader035.vdocument.in/reader035/viewer/2022062514/558e94cb1a28ab3b108b4722/html5/thumbnails/37.jpg)
Σi=1
Theorem:
The Harmonic Series and the Integral Test
(p-series)
CGs if and only if p > 1. ∞
np1
Proof:
Σi=1
CGs if and only if CGs. ∞
np1
By the integral test,
∫1 xp
1∞
dx
![Page 38: 25 the harmonic series and the integral test](https://reader035.vdocument.in/reader035/viewer/2022062514/558e94cb1a28ab3b108b4722/html5/thumbnails/38.jpg)
Σi=1
Theorem:
The Harmonic Series and the Integral Test
(p-series)
CGs if and only if p > 1. ∞
np1
Proof: By the integral test,
By the p-theorem, this integral CGs if and only if p >1.
Σi=1
CGs if and only if CGs. ∞
np1
∫1 xp
1∞
dx
![Page 39: 25 the harmonic series and the integral test](https://reader035.vdocument.in/reader035/viewer/2022062514/558e94cb1a28ab3b108b4722/html5/thumbnails/39.jpg)
Σi=1
Theorem:
The Harmonic Series and the Integral Test
(p-series)
CGs if and only if p > 1. ∞
np1
Proof: By the integral test,
By the p-theorem, this integral CGs if and only if p >1.
So CGs if and only if p > 1. Σi=1
∞
np1
Σi=1
CGs if and only if CGs. ∞
np1
∫1 xp
1∞
dx
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Σi=1
Theorem:
The Harmonic Series and the Integral Test
(p-series)
CGs if and only if p > 1. ∞
np1
Proof: By the integral test,
By the p-theorem, this integral CGs if and only if p >1.
So CGs if and only if p > 1.
Example:
a. Σi=1
∞
n3/21
b. Σi=1
∞
n1
Σi=1
∞
np1
Σi=1
CGs if and only if CGs. ∞
np1
∫1 xp
1∞
dx
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Σi=1
Theorem:
The Harmonic Series and the Integral Test
(p-series)
CGs if and only if p > 1. ∞
np1
Proof: By the integral test,
By the p-theorem, this integral CGs if and only if p >1.
So CGs if and only if p > 1.
Example:
a. CGs since 3/2 > 1.Σi=1
∞
n3/21
b. Σi=1
∞
n1
Σi=1
∞
np1
Σi=1
CGs if and only if CGs. ∞
np1
∫1 xp
1∞
dx
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Σi=1
Theorem:
The Harmonic Series and the Integral Test
(p-series)
CGs if and only if p > 1. ∞
np1
Proof: By the integral test,
By the p-theorem, this integral CGs if and only if p >1.
So CGs if and only if p > 1.
Example:
a. CGs since 3/2 > 1.Σi=1
∞
n3/21
b. DGs since 1/2 < 1.Σi=1
∞
n1
Σi=1
∞
np1
Σi=1
CGs if and only if CGs. ∞
np1
∫1 xp
1∞
dx
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Σi=1
Theorem:
The Harmonic Series and the Integral Test
(p-series)
CGs if and only if p > 1. ∞
np1
Proof: By the integral test,
By the p-theorem, this integral CGs if and only if p >1.
So CGs if and only if p > 1.
Example:
a. CGs since 3/2 > 1.Σi=1
∞
n3/21
b. DGs since 1/2 < 1.Σi=1
∞
n1
This theorem applies to series that are p-series except for finitely many terms (eventual p-series).
Σi=1
∞
np1
Σi=1
CGs if and only if CGs. ∞
np1
∫1 xp
1∞
dx
Recall the following theorems of improper integrals.
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(The Floor Theorem)
The Harmonic Series and the Integral Test
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(The Floor Theorem)
y = f(x)
y = g(x)∞
The Harmonic Series and the Integral Test
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(The Floor Theorem) If f(x) > g(x) > 0 and g(x) dx = ∞, ∫
a
b
y = f(x)
y = g(x)∞
The Harmonic Series and the Integral Test
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(The Floor Theorem) If f(x) > g(x) > 0 and g(x) dx = ∞, then f(x) = ∞. ∫
a
b
∫a
b
y = f(x)
y = g(x)∞
The Harmonic Series and the Integral Test
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(The Floor Theorem) If f(x) > g(x) > 0 and g(x) dx = ∞, then f(x) = ∞. ∫
a
b
∫a
b
y = f(x)
y = g(x)∞
(The Ceiling theorem)
The Harmonic Series and the Integral Test
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(The Floor Theorem) If f(x) > g(x) > 0 and g(x) dx = ∞, then f(x) = ∞. ∫
a
b
∫a
b
y = f(x)
y = g(x)∞
(The Ceiling theorem)
y = f(x)
y = g(x)
N
The Harmonic Series and the Integral Test
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(The Floor Theorem) If f(x) > g(x) > 0 and g(x) dx = ∞, then f(x) = ∞. ∫
a
b
∫a
b
y = f(x)
y = g(x)∞
(The Ceiling theorem)
If f(x) > g(x) > 0 and f(x) dx = N converges ∫a
b
y = f(x)
y = g(x)
N
The Harmonic Series and the Integral Test
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(The Floor Theorem) If f(x) > g(x) > 0 and g(x) dx = ∞, then f(x) = ∞. ∫
a
b
∫a
b
y = f(x)
y = g(x)∞
(The Ceiling theorem)
If f(x) > g(x) > 0 and f(x) dx = N converges then
g(x) dx converges also.∫a
b
∫a
b
y = f(x)
y = g(x)
N
The Harmonic Series and the Integral Test
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The Harmonic Series and the Integral TestBy the same logic we have their discrete versions.
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The Harmonic Series and the Integral TestBy the same logic we have their discrete versions.The Floor Theorem
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The Harmonic Series and the Integral TestBy the same logic we have their discrete versions.The Floor TheoremLet {an} and {bn} be two sequences and an > bn > 0.
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The Harmonic Series and the Integral Test
Suppose bn = ∞, then an = ∞. Σi=k
∞
Σi=k
∞
By the same logic we have their discrete versions.The Floor TheoremLet {an} and {bn} be two sequences and an > bn > 0.
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The Harmonic Series and the Integral Test
Suppose bn = ∞, then an = ∞. Σi=k
∞
Σi=k
∞
Example: Does CG or DG?
By the same logic we have their discrete versions.
Σi=2
∞
Ln(n)
1
The Floor TheoremLet {an} and {bn} be two sequences and an > bn > 0.
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The Harmonic Series and the Integral Test
Suppose bn = ∞, then an = ∞. Σi=k
∞
Σi=k
∞
Example: Does CG or DG?
For n > 1, n > Ln(n), (why?)
By the same logic we have their discrete versions.
Σi=2
∞
Ln(n)
1
The Floor TheoremLet {an} and {bn} be two sequences and an > bn > 0.
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The Harmonic Series and the Integral Test
Suppose bn = ∞, then an = ∞. Σi=k
∞
Σi=k
∞
Example: Does CG or DG?
Ln(n) 1 > n .
1For n > 1, n > Ln(n), (why?)
By the same logic we have their discrete versions.
Σi=2
∞
Ln(n)
1
so
The Floor TheoremLet {an} and {bn} be two sequences and an > bn > 0.
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The Harmonic Series and the Integral Test
Suppose bn = ∞, then an = ∞. Σi=k
∞
Σi=k
∞
Example: Does CG or DG?
Ln(n) 1 > n .
1For n > 1, n > Ln(n), (why?)
Σi=2 n
1Hence Σi=2 Ln(n)
2
By the same logic we have their discrete versions.
>
Σi=2
∞
Ln(n)
1
so
The Floor TheoremLet {an} and {bn} be two sequences and an > bn > 0.
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The Harmonic Series and the Integral Test
Suppose bn = ∞, then an = ∞. Σi=k
∞
Σi=k
∞
Example: Does CG or DG?
Ln(n) 1 > n .
1For n > 1, n > Ln(n), (why?)
Σi=2 n
1Hence Σi=2 Ln(n)
2
By the same logic we have their discrete versions.
> = ∞ because it’s harmonic.
Σi=2
∞
Ln(n)
1
so
The Floor TheoremLet {an} and {bn} be two sequences and an > bn > 0.
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The Harmonic Series and the Integral Test
Suppose bn = ∞, then an = ∞. Σi=k
∞
Σi=k
∞
Example: Does CG or DG?
Ln(n) 1 > n .
1For n > 1, n > Ln(n), (why?)
Σi=2 n
1
Therefore
Hence Σi=2 Ln(n)
2
By the same logic we have their discrete versions.
> = ∞ because it’s harmonic.
Σi=2
∞
Ln(n)
1
Σi=2 Ln(n)
2
so
= ∞ or that it DGs.
The Floor TheoremLet {an} and {bn} be two sequences and an > bn > 0.
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The Harmonic Series and the Integral Test
Suppose bn = ∞, then an = ∞. Σi=k
∞
Σi=k
∞
Example: Does CG or DG?
Ln(n) 1 > n .
1For n > 1, n > Ln(n), (why?)
Σi=2 n
1
Therefore
Hence Σi=2 Ln(n)
2
By the same logic we have their discrete versions.
> = ∞ because it’s harmonic.
Σi=2
∞
Ln(n)
1
Σi=2 Ln(n)
2
so
= ∞ or that it DGs.
The Floor TheoremLet {an} and {bn} be two sequences and an > bn > 0.
Note that no conclusion can be drawn about Σan if that Σ bn < ∞ i.e. Σ an may CG or it may DG. (Why so?)
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The Harmonic Series and the Integral TestThe Ceiling Theorem
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The Harmonic Series and the Integral TestThe Ceiling TheoremLet {an} and {bn} be two sequences and an > bn > 0.
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The Harmonic Series and the Integral Test
Suppose that an CGs, then bn CGs. Σi=k
Σi=k
The Ceiling TheoremLet {an} and {bn} be two sequences and an > bn > 0.
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The Harmonic Series and the Integral Test
Suppose that an CGs, then bn CGs. Σi=k
Σi=k
Example: Does CG or DG?
The Ceiling TheoremLet {an} and {bn} be two sequences and an > bn > 0.
Σi=1
∞
n2 + 4
2
2 2
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The Harmonic Series and the Integral Test
Suppose that an CGs, then bn CGs. Σi=k
Σi=k
Example: Does CG or DG?
The Ceiling TheoremLet {an} and {bn} be two sequences and an > bn > 0.
Σi=1
∞
n2 + 4
2
Compare with n2 + 4
2n2 2
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The Harmonic Series and the Integral Test
Suppose that an CGs, then bn CGs. Σi=k
Σi=k
Example: Does CG or DG?
n2 + 4 2>n2
2. we have
The Ceiling TheoremLet {an} and {bn} be two sequences and an > bn > 0.
Σi=1
∞
n2 + 4
2
Compare with n2 + 4
2n2 2
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The Harmonic Series and the Integral Test
Suppose that an CGs, then bn CGs. Σi=k
Σi=k
Example: Does CG or DG?
n2 + 4 2>n2
2.
Σ n2 2
we have
The Ceiling TheoremLet {an} and {bn} be two sequences and an > bn > 0.
Σi=1
∞
n2 + 4
2
Compare with n2 + 4
2n2 2
= 2Σ n2 1
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The Harmonic Series and the Integral Test
Suppose that an CGs, then bn CGs. Σi=k
Σi=k
Example: Does CG or DG?
n2 + 4 2>n2
2.
Σ n2 2
we have
CGs since it’s the p–series with p = 2 > 1,
The Ceiling TheoremLet {an} and {bn} be two sequences and an > bn > 0.
Σi=1
∞
n2 + 4
2
Compare with n2 + 4
2n2 2
= 2Σ n2 1
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The Harmonic Series and the Integral Test
Suppose that an CGs, then bn CGs. Σi=k
Σi=k
Example: Does CG or DG?
n2 + 4 2>n2
2.
Σ n2 2
we have
CGs since it’s the p–series with p = 2 > 1,
n2 + 1
The Ceiling TheoremLet {an} and {bn} be two sequences and an > bn > 0.
Σi=1
∞
n2 + 4
2
Compare with n2 + 4
2n2 2
2we see that Σ CGs also.
= 2Σ n2 1
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The Harmonic Series and the Integral Test
Suppose that an CGs, then bn CGs. Σi=k
Σi=k
Example: Does CG or DG?
n2 + 4 2>n2
2.
Σ n2 2
we have
CGs since it’s the p–series with p = 2 > 1,
n2 + 1
The Ceiling TheoremLet {an} and {bn} be two sequences and an > bn > 0.
Σi=1
∞
n2 + 4
2
Compare with n2 + 4
2n2 2
2
Note that no conclusion can be drawn about Σbn if that Σan = ∞ i.e. Σ bn may CG or it may DG. (Why so?)
we see that Σ CGs also.
= 2Σ n2 1