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

∞