Transcript
Infinite Series
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 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,
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.
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.
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.
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.
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.
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.
Two main questions concerning a series are* does the series CG or DG?* if it converges, what is the sum?
Infinite Series
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.
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.
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
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
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
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
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
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
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
Infinite SeriesFormula for Geometric Series:
Σn = 0
∞arn
where -1 < r < 1. ( | r | < 1 )
= a + ar + ar2 + ar3… = a 1 – r
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
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 + .. =
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 + … )
∞
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 =
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
∞
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.
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,
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∞
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.
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.
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.
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.
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.
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.
Infinite Series
Example: Find the sum Σn=1
∞ (-2)n+2*5
3n-1*7
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. ∞
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
∞
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
∞
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
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
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 (
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
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
Infinite SeriesII. The Telescoping Series:
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
∞
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
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
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
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 ) …
∞
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.
∞
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
∞
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