24 infinite series
description
Transcript of 24 infinite series
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Infinite Series
![Page 2: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/2.jpg)
We want to define the sum of infinitely many terms
a1 + a2 + a3 + .. =
which is called an (infinite) series.
Infinite Series
Σi = 1
∞
ai
![Page 3: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/3.jpg)
We want to define the sum of infinitely many terms
a1 + a2 + a3 + .. =
which is called an (infinite) series.
Infinite Series
Σi = 1
∞
ai
We define the n'th partial sum sn of a series to bethe sum of the first n terms of the sequence,
![Page 4: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/4.jpg)
We want to define the sum of infinitely many terms
a1 + a2 + a3 + .. =
which is called an (infinite) series.
Infinite Series
Σi = 1
∞
ai
We define the n'th partial sum sn of a series to bethe sum of the first n terms of the sequence, i.e. sn = a1 + a2 + a3 + … + an.
![Page 5: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/5.jpg)
We want to define the sum of infinitely many terms
a1 + a2 + a3 + .. =
which is called an (infinite) series.
Infinite Series
Σi = 1
∞
ai
So s1 = a1, s2 = a1 + a2, s3 = a1 + a2 + a3, etc..
We define the n'th partial sum sn of a series to bethe sum of the first n terms of the sequence, i.e. sn = a1 + a2 + a3 + … + an.
![Page 6: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/6.jpg)
We want to define the sum of infinitely many terms
a1 + a2 + a3 + .. =
which is called an (infinite) series.
Infinite Series
Σi = 1
∞
ai
So s1 = a1, s2 = a1 + a2, s3 = a1 + a2 + a3, etc..s1, s2 , s3, … form a sequence.
We define the n'th partial sum sn of a series to bethe sum of the first n terms of the sequence, i.e. sn = a1 + a2 + a3 + … + an.
![Page 7: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/7.jpg)
We want to define the sum of infinitely many terms
a1 + a2 + a3 + .. =
which is called an (infinite) series.
Infinite Series
Σi = 1
∞
ai
So s1 = a1, s2 = a1 + a2, s3 = a1 + a2 + a3, etc..s1, s2 , s3, … form a sequence. We define the limit of {sn} to be the sum of the series,
We define the n'th partial sum sn of a series to bethe sum of the first n terms of the sequence, i.e. sn = a1 + a2 + a3 + … + an.
![Page 8: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/8.jpg)
We want to define the sum of infinitely many terms
a1 + a2 + a3 + .. =
which is called an (infinite) series.
Infinite Series
Σi = 1
∞
ai
So s1 = a1, s2 = a1 + a2, s3 = a1 + a2 + a3, etc..s1, s2 , s3, … form a sequence. We define the limit of {sn} to be the sum of the series,
i.e. lim sn = as n ∞. Σi = 1
∞ai
We define the n'th partial sum sn of a series to bethe sum of the first n terms of the sequence, i.e. sn = a1 + a2 + a3 + … + an.
![Page 9: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/9.jpg)
We want to define the sum of infinitely many terms
a1 + a2 + a3 + .. =
which is called an (infinite) series.
Infinite Series
Σi = 1
∞
ai
So s1 = a1, s2 = a1 + a2, s3 = a1 + a2 + a3, etc..s1, s2 , s3, … form a sequence. We define the limit of {sn} to be the sum of the series,
i.e. lim sn = as n ∞.
We say the series converges if {sn} converges (CG) and that it diverges (DG) if {sn} diverges.
Σi = 1
∞ai
We define the n'th partial sum sn of a series to bethe sum of the first n terms of the sequence, i.e. sn = a1 + a2 + a3 + … + an.
![Page 10: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/10.jpg)
Two main questions concerning a series are* does the series CG or DG?* if it converges, what is the sum?
Infinite Series
![Page 11: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/11.jpg)
Two main questions concerning a series are* does the series CG or DG?* if it converges, what is the sum?
Infinite Series
These are not easy problems for most series.
![Page 12: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/12.jpg)
Two main questions concerning a series are* does the series CG or DG?* if it converges, what is the sum?
Infinite Series
These are not easy problems for most series. But there are two types of series that converge and their sum may be calculate with elementary algebra.
![Page 13: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/13.jpg)
Two main questions concerning a series are* does the series CG or DG?* if it converges, what is the sum?
Infinite Series
I. Geometric Series: Σn=0
∞arn = a + ar + ar2 + ar3…
These are not easy problems for most series. But there are two types of series that converge and their sum may be calculate with elementary algebra.
|r| < 1
![Page 14: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/14.jpg)
Two main questions concerning a series are* does the series CG or DG?* if it converges, what is the sum?
Infinite Series
I. Geometric Series:
We observe the algebraic patterns:(1 – r)(1 + r + r2 … + rn-1) = 1 – rn
,
Σn=0
∞arn = a + ar + ar2 + ar3…
These are not easy problems for most series. But there are two types of series that converge and their sum may be calculate with elementary algebra.
|r| < 1
![Page 15: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/15.jpg)
Two main questions concerning a series are* does the series CG or DG?* if it converges, what is the sum?
Infinite Series
I. Geometric Series:
We observe the algebraic patterns:(1 – r)(1 + r + r2 … + rn-1) = 1 – rn
, hence
1 + r + r2 … + rn-1 = 1 – rn 1 – r
Σn=0
∞arn = a + ar + ar2 + ar3…
These are not easy problems for most series. But there are two types of series that converge and their sum may be calculate with elementary algebra.
|r| < 1
![Page 16: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/16.jpg)
Two main questions concerning a series are* does the series CG or DG?* if it converges, what is the sum?
Infinite Series
I. Geometric Series:
We observe the algebraic patterns:(1 – r)(1 + r + r2 … + rn-1) = 1 – rn
, hence
1 + r + r2 … + rn-1 = 1 – rn 1 – r
As n∞, rn 0 if | r | < 1,
Σn=0
∞arn = a + ar + ar2 + ar3…
These are not easy problems for most series. But there are two types of series that converge and their sum may be calculate with elementary algebra.
|r| < 1
![Page 17: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/17.jpg)
Two main questions concerning a series are* does the series CG or DG?* if it converges, what is the sum?
Infinite Series
I. Geometric Series:
We observe the algebraic patterns:(1 – r)(1 + r + r2 … + rn-1) = 1 – rn
, hence
1 + r + r2 … + rn-1 = 1 – rn 1 – r
As n∞, rn 0 if | r | < 1, then
lim (1 + r + r2 … + rn-1) Σ∞
rn =n∞
1 – rn
1 – r =
Σn=0
∞arn = a + ar + ar2 + ar3…
n=0lim n∞
These are not easy problems for most series. But there are two types of series that converge and their sum may be calculate with elementary algebra.
|r| < 1
![Page 18: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/18.jpg)
Two main questions concerning a series are* does the series CG or DG?* if it converges, what is the sum?
Infinite Series
I. Geometric Series:
We observe the algebraic patterns:(1 – r)(1 + r + r2 … + rn-1) = 1 – rn
, hence
1 + r + r2 … + rn-1 = 1 – rn 1 – r
As n∞, rn 0 if | r | < 1, then
lim (1 + r + r2 … + rn-1) Σ∞
rn =n∞
1 – rn
1 – r =
Σn=0
∞arn = a + ar + ar2 + ar3…
n=0lim n∞
These are not easy problems for most series. But there are two types of series that converge and their sum may be calculate with elementary algebra.
|r| < 1
0
![Page 19: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/19.jpg)
Two main questions concerning a series are* does the series CG or DG?* if it converges, what is the sum?
Infinite Series
I. Geometric Series:
We observe the algebraic patterns:(1 – r)(1 + r + r2 … + rn-1) = 1 – rn
, hence
1 + r + r2 … + rn-1 = 1 – rn 1 – r
As n∞, rn 0 if | r | < 1, then
lim (1 + r + r2 … + rn-1) Σ∞
rn =n∞
1 – rn
1 – r =
Σn=0
∞arn = a + ar + ar2 + ar3…
n=01
1 – r =lim
n∞
These are not easy problems for most series. But there are two types of series that converge and their sum may be calculate with elementary algebra.
|r| < 1
0
![Page 20: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/20.jpg)
Infinite SeriesFormula for Geometric Series:
Σn = 0
∞arn
where -1 < r < 1. ( | r | < 1 )
= a + ar + ar2 + ar3… = a 1 – r
![Page 21: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/21.jpg)
Infinite Series
Example: Find the sum Σn=1
∞5 3n
Formula for Geometric Series:
Σn = 0
∞arn
where -1 < r < 1. ( | r | < 1 )
= a + ar + ar2 + ar3… = a 1 – r
![Page 22: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/22.jpg)
Infinite Series
Example: Find the sum Σn=1
∞
1st: In the expanded form.
5 3n
Formula for Geometric Series:
Σn = 0
∞arn
where -1 < r < 1. ( | r | < 1 )
= a + ar + ar2 + ar3… = a 1 – r
Σn=1
∞5 3n = 5
3 + 5 32 + 5
33 + .. =
![Page 23: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/23.jpg)
Infinite Series
Example: Find the sum Σn=1
1st: In the expanded form.
5 3n
Formula for Geometric Series:
Σn = 0
∞arn
where -1 < r < 1. ( | r | < 1 )
= a + ar + ar2 + ar3… = a 1 – r
Σn=1
∞5 3n = 5
3 + 5 32 + 5
33 + .. = 5 3
(1 + 1 3 + 1
32 + … )
∞
![Page 24: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/24.jpg)
Infinite Series
Example: Find the sum Σn=1
∞
1st: In the expanded form.
1 1 – 1/3
5 3n
Formula for Geometric Series:
Σn = 0
∞arn
where -1 < r < 1. ( | r | < 1 )
= a + ar + ar2 + ar3… = a 1 – r
Σn=1
∞5 3n = 5
3 + 5 32 + 5
33 + .. = 5 3
(1 + 1 3 + 1
32 + … )
= 5 3 =
![Page 25: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/25.jpg)
Infinite Series
Example: Find the sum Σn=1
1st: In the expanded form.
1 1 – 1/3
5 3n
Formula for Geometric Series:
Σn = 0
∞arn
where -1 < r < 1. ( | r | < 1 )
= a + ar + ar2 + ar3… = a 1 – r
Σn=1
∞5 3n = 5
3 + 5 32 + 5
33 + .. = 5 3
(1 + 1 3 + 1
32 + … )
= 5 3
3 2
= 5 3 = 5
2
∞
![Page 26: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/26.jpg)
Infinite Series
Example: Find the sum Σn=1
∞
1st: In the expanded form.
1 1 – 1/3
5 3n
Formula for Geometric Series:
Σn = 0
∞arn
where -1 < r < 1. ( | r | < 1 )
= a + ar + ar2 + ar3… = a 1 – r
Σn=1
∞5 3n = 5
3 + 5 32 + 5
33 + .. = 5 3
(1 + 1 3 + 1
32 + … )
= 5 3
3 2
= 5 3 = 5
2
2nd: By shifting the index.
![Page 27: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/27.jpg)
Infinite Series
Example: Find the sum Σn=1
∞
1st: In the expanded form.
1 1 – 1/3
5 3n
Formula for Geometric Series:
Σn = 0
∞arn
where -1 < r < 1. ( | r | < 1 )
= a + ar + ar2 + ar3… = a 1 – r
Σn=1
∞5 3n = 5
3 + 5 32 + 5
33 + .. = 5 3
(1 + 1 3 + 1
32 + … )
= 5 3
3 2
= 5 3 = 5
2
2nd: By shifting the index.
Set k=n–1,
![Page 28: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/28.jpg)
Infinite Series
Example: Find the sum Σn=1
∞
1st: In the expanded form.
1 1 – 1/3
5 3n
Formula for Geometric Series:
Σn = 0
∞arn
where -1 < r < 1. ( | r | < 1 )
= a + ar + ar2 + ar3… = a 1 – r
Σn=1
∞5 3n = 5
3 + 5 32 + 5
33 + .. = 5 3
(1 + 1 3 + 1
32 + … )
= 5 3
3 2
= 5 3 = 5
2
2nd: By shifting the index.
Set k=n–1, as n goes from 1∞, k goes from 0∞
![Page 29: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/29.jpg)
Infinite Series
Example: Find the sum Σn=1
∞
1st: In the expanded form.
1 1 – 1/3
5 3n
Formula for Geometric Series:
Σn = 0
∞arn
where -1 < r < 1. ( | r | < 1 )
= a + ar + ar2 + ar3… = a 1 – r
Σn=1
∞5 3n = 5
3 + 5 32 + 5
33 + .. = 5 3
(1 + 1 3 + 1
32 + … )
= 5 3
3 2
= 5 3 = 5
2
2nd: By shifting the index.
Set k=n–1, as n goes from 1∞, k goes from 0∞ and n=k+1.
![Page 30: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/30.jpg)
Infinite Series
Example: Find the sum Σn=1
∞
1st: In the expanded form.
1 1 – 1/3
5 3n
Formula for Geometric Series:
Σn = 0
∞arn
where -1 < r < 1. ( | r | < 1 )
= a + ar + ar2 + ar3… = a 1 – r
Σn=1
∞5 3n = 5
3 + 5 32 + 5
33 + .. = 5 3
(1 + 1 3 + 1
32 + … )
= 5 3
3 2
= 5 3 = 5
2
2nd: By shifting the index.
Σn=1
5 3n =
∞
Σk=0
5 3k+1
∞
Set k=n–1, as n goes from 1∞, k goes from 0∞ and n=k+1.
![Page 31: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/31.jpg)
Infinite Series
Example: Find the sum Σn=1
∞
1st: In the expanded form.
1 1 – 1/3
5 3n
Formula for Geometric Series:
Σn = 0
∞arn
where -1 < r < 1. ( | r | < 1 )
= a + ar + ar2 + ar3… = a 1 – r
Σn=1
∞5 3n = 5
3 + 5 32 + 5
33 + .. = 5 3
(1 + 1 3 + 1
32 + … )
= 5 3
3 2
= 5 3 = 5
2
2nd: By shifting the index.
Σn=1
5 3n =
∞
Σk=0
5 3k+1 =
∞
Σk=0
5 ∞
3*3k
Set k=n–1, as n goes from 1∞, k goes from 0∞ and n=k+1.
![Page 32: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/32.jpg)
Infinite Series
Example: Find the sum Σn=1
∞
1st: In the expanded form.
1 1 – 1/3
5 3n
Formula for Geometric Series:
Σn = 0
∞arn
where -1 < r < 1. ( | r | < 1 )
= a + ar + ar2 + ar3… = a 1 – r
Σn=1
∞5 3n = 5
3 + 5 32 + 5
33 + .. = 5 3
(1 + 1 3 + 1
32 + … )
= 5 3
3 2
= 5 3 = 5
2
2nd: By shifting the index.
Σn=1
5 3n =
∞
Σk=0
5 3k+1 =
∞
Σk=0
5 ∞
3*3k = Σk=0
1
∞
3k 5 3
Set k=n–1, as n goes from 1∞, k goes from 0∞ and n=k+1.
![Page 33: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/33.jpg)
Infinite Series
Example: Find the sum Σn=1
∞
1st: In the expanded form.
1 1 – 1/3
5 3n
Formula for Geometric Series:
Σn = 0
∞arn
where -1 < r < 1. ( | r | < 1 )
= a + ar + ar2 + ar3… = a 1 – r
Σn=1
∞5 3n = 5
3 + 5 32 + 5
33 + .. = 5 3
(1 + 1 3 + 1
32 + … )
= 5 3
3 2
= 5 3 = 5
2
2nd: By shifting the index.
Σn=1
5 3n =
∞
Σk=0
5 3k+1 =
∞
Σk=0
5 ∞
3*3k = Σk=0
1
∞
3k 5 3 =
5 3
1 1 – 1/3
Set k=n–1, as n goes from 1∞, k goes from 0∞ and n=k+1.
![Page 34: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/34.jpg)
Infinite Series
Example: Find the sum Σn=1
∞
1st: In the expanded form.
1 1 – 1/3
5 3n
Formula for Geometric Series:
Σn = 0
∞arn
where -1 < r < 1. ( | r | < 1 )
= a + ar + ar2 + ar3… = a 1 – r
Σn=1
∞5 3n = 5
3 + 5 32 + 5
33 + .. = 5 3
(1 + 1 3 + 1
32 + … )
= 5 3
3 2
= 5 3 = 5
2
2nd: By shifting the index.
Σn=1
5 3n =
∞
Σk=0
5 3k+1 =
∞
Σk=0
5 ∞
3*3k = Σk=0
1
∞
3k 5 3 =
5 3
1 1 – 1/3 =
5 2
Set k=n–1, as n goes from 1∞, k goes from 0∞ and n=k+1.
![Page 35: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/35.jpg)
Infinite Series
Example: Find the sum Σn=1
∞ (-2)n+2*5
3n-1*7
![Page 36: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/36.jpg)
Infinite Series
Example: Find the sum Σn=1
∞ (-2)n+2*5
3n-1*7
Set k = n – 1 so k goes from 0 and n = k +1. ∞
![Page 37: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/37.jpg)
Infinite Series
Example: Find the sum Σn=1
∞ (-2)n+2*5
3n-1*7
Set k = n – 1 so k goes from 0 and n = k +1.
Hence Σn=1
∞ (-2)n+2*5
3n-1*7
∞
![Page 38: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/38.jpg)
Infinite Series
Example: Find the sum Σn=1
∞ (-2)n+2*5
3n-1*7
Set k = n – 1 so k goes from 0 and n = k +1.
Hence Σn=1
∞ (-2)n+2*5
3n-1*7
= Σk=0
∞ (-2)k+1+2*5
3k+1-1*7
∞
![Page 39: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/39.jpg)
Infinite Series
Example: Find the sum Σn=1
∞ (-2)n+2*5
3n-1*7
Set k = n – 1 so k goes from 0 and n = k +1.
Hence Σn=1
∞ (-2)n+2*5
3n-1*7
= Σk=0
∞ (-2)k+1+2*5
3k+1-1*7
∞
= Σk=0
∞ (-2)k+3*5
3k*7
![Page 40: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/40.jpg)
Infinite Series
Example: Find the sum Σn=1
∞ (-2)n+2*5
3n-1*7
Set k = n – 1 so k goes from 0 and n = k +1.
Hence Σn=1
∞ (-2)n+2*5
3n-1*7
= Σk=0
∞ (-2)k+1+2*5
3k+1-1*7
∞
= Σk=0
∞ (-2)k+3*5
3k*7
= Σk=0
∞ (-2)k(-8*5) 3k*7
![Page 41: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/41.jpg)
Infinite Series
Example: Find the sum Σn=1
∞ (-2)n+2*5
3n-1*7
Set k = n – 1 so k goes from 0 and n = k +1.
Hence Σn=1
∞ (-2)n+2*5
3n-1*7
= Σk=0
∞ (-2)k+1+2*5
3k+1-1*7
∞
= Σk=0
∞ (-2)k+3*5
3k*7
= Σk=0
∞ (-2)k(-8*5) 3k*7
= Σk=0
∞ -2 3
-40 7 )k (
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Infinite Series
Example: Find the sum Σn=1
∞ (-2)n+2*5
3n-1*7
Set k = n – 1 so k goes from 0 and n = k +1.
Hence Σn=1
∞ (-2)n+2*5
3n-1*7
= Σk=0
∞ (-2)k+1+2*5
3k+1-1*7
∞
= Σk=0
∞ (-2)k+3*5
3k*7
= Σk=0
∞ (-2)k(-8*5) 3k*7
= Σk=0
∞ -2 3
-40 7 )k ( = -40
7 1
1 + 2/3
![Page 43: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/43.jpg)
Infinite Series
Example: Find the sum Σn=1
∞ (-2)n+2*5
3n-1*7
Set k = n – 1 so k goes from 0 and n = k +1.
Hence Σn=1
∞ (-2)n+2*5
3n-1*7
= Σk=0
∞ (-2)k+1+2*5
3k+1-1*7
∞
= Σk=0
∞ (-2)k+3*5
3k*7
= Σk=0
∞ (-2)k(-8*5) 3k*7
= Σk=0
∞ -2 3
-40 7 )k ( = -40
7 1
1 + 2/3 = -40 7
3 5 = -24
7
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Infinite SeriesII. The Telescoping Series:
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Infinite Series
The series Σ r n + p
II. The Telescoping Series:
– r n + q ][
constants called telescoping series.
where p, q, and r are n=1
∞
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Infinite Series
The series Σn=1
∞r
n + p
II. The Telescoping Series:
– r n + q ][
constants called telescoping series. They are so named due to the cancelation of the terms in the sum.
where p, q, and r are
![Page 47: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/47.jpg)
Infinite Series
The series Σn=1
∞r
n + p
II. The Telescoping Series:
– r n + q ][
constants called telescoping series. They are so named due to the cancelation of the terms in the sum.
Example: Find Σn=1
∞1 n + 2
– 1 n + 4 ][
where p, q, and r are
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Infinite Series
The series Σn=1
∞r
n + p
II. The Telescoping Series:
– r n + q ][
constants called telescoping series. They are so named due to the cancelation of the terms in the sum.
Example: Find Σn=1
∞1 n + 2
– 1 n + 4 ][
Σn=1
∞1 n + 2
– 1 n + 4 ][
=1 3 +( – 1
5 )
1 4 ( – 1
6 )+
1 5
( – 1 7
)+1 6 ( – 1
8 )+
1 7 ( – 1
9 ) …
where p, q, and r are
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Infinite Series
The series Σn=1
∞r
n + p
II. The Telescoping Series:
– r n + q ][
constants called telescoping series. They are so named due to the cancelation of the terms in the sum.
where p, q, and r are
Example: Find Σn=1
1 n + 2
– 1 n + 4 ][
Σn=1
∞1 n + 2
– 1 n + 4 ][
=1 3 +( – 1
5 )
1 4 ( – 1
6 )+
1 5
( – 1 7
)+1 6 ( – 1
8 )+
1 7 ( – 1
9 ) …
∞
![Page 50: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/50.jpg)
Infinite Series
The series Σn=1
∞r
n + p
II. The Telescoping Series:
– r n + q ][
constants called telescoping series. They are so named due to the cancelation of the terms in the sum.
where p, q, and r are
Example: Find Σn=1
1 n + 2
– 1 n + 4 ][
Σn=1
∞1 n + 2
– 1 n + 4 ][
=1 3 +( – 1
5 )
1 4 ( – 1
6 )+
1 5
( – 1 7
)+1 6 ( – 1
8 )+
1 7 ( – 1
9 ) …
= 1 3
+ 1 4 = 7
12.
∞
![Page 51: 24 infinite series](https://reader031.fdocuments.net/reader031/viewer/2022013121/546ac038af795979438b55c4/html5/thumbnails/51.jpg)
Infinite Series
The series Σn=1
∞r
n + p
II. The Telescoping Series:
– r n + q ][
constants called telescoping series. They are so named due to the cancelation of the terms in the sum.
where p, q, and r are
Example: Find Σn=1
1 n + 2
– 1 n + 4 ][
Σn=1
∞1 n + 2
– 1 n + 4 ][
=1 3 +( – 1
5 )
1 4 ( – 1
6 )+
1 5
( – 1 7
)+1 6 ( – 1
8 )+
1 7 ( – 1
9 ) …
= 1 3
+ 1 4 = 7
12. Note that if x2 + bx + c is factorable, then
1 x2+bx+c =
r n + p
– r n + q
∞