For student 1 (Sequences and SERIES )

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A sequence is an infinite ordered list of numbers .

Example

The sequence of odd positive integers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29 . . .

Symbolically: a1, a2, a3, a4, a5, a6, a7, . . . , an, . . .= {an} or {

an

}

n1

The nth term (general formula) definition of a sequence:

Some sequences can be defined with a general formula in term of the index n.

Example

The sequence 1, 3, 5, 7, . . . has the formula

a

n

= ………..

A recursive definition of a sequence:

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The limit of a sequence is the value to which its terms approach indefinitely as n becomes large.

We write that the limit of a sequence a

n

is L in the following way:

  n n a

lim

L or

a

n

l

as n→∞.

 

n

n

1

lim

 

n

n

n

1

lim

  2

lim

n

n

Operations with Limits

If a

n

→ a and b

n

→ b then:

(a

n

+ b

n

) → a + b.

(a

n

− b

n

) → a − b.

c.a

n

→ ca for any constant c.

a

n

b

n

→ ab.

b a b a

n

n

if b ≠ 0.

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Example: Evaluate the limit, if it exists.

1-3

9

6

3

lim

2 2

 

n

n

n

n

2-3

5

9

6

lim

3 2

 

n

n

n

n

3-3

9

6

2

lim

2

 

n

n

n

n

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Convergent limit: If a sequence has a (finite) limit

Divergent limit: If a sequence becomes an arbitrary large or no end

Example: Using L'Hopital's Rule

Determine if the sequence

n

n

a

n

ln

is convergence or divergence

Example

Find

  

n

n n

n

a

lim

r

lim

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A sequence is:

-

increasing if a

n+1

≥ a

n

for every n.

-

strictly increasing if a

n+1

> a

n

for every n.

-

decreasing if a

n+1

≤ a

n

for every n.

-

strictly decreasing if a

n+1

< a

n

for every n.

It is called monotonic if it is either increasing or decreasing.

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Example:

Prove that the sequence

n

n

a

n

1

is strictly decreasing.

Prove that the sequence

1

n

n

a

n

is strictly increasing.

Classify the following sequence:

1-n

a

n

1

1

2-

(

1

2

)

n

a

n

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Bounded

A sequence is

-

bounded above if there is a number M such that a

n

< M for all n.

-

bounded below if there is a number m such that m < a

n

for all n.

-

It is called just bounded if it is bounded above and below.

Example:

Prove that the sequence

n

n

a

n

1

is bounded

Example:

Prove that the sequence

n

n

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A series is an infinite sum:

...

a

+

·

·

·

+

a

+

a

+

a

=

a

1 n 3 2 1 n

n i

In order to define the value of these sum we start be defining its sequence of partial sums

partial sums

n i n

S

1 n 3 2 1

n

=

a

+

a

+

a

+

·

·

·

+

a

a

1

S

2

S

. . 3  S

.

Then, if

Sn S

nlim 

exists the series is called convergent and its sum is that limit:

  

1

lim

S

=

an

n n n

S

Otherwise the series is called divergent.

a.

Find the nth partial sum of the following series

 1 n

2

1

n

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b.

Telescopic Series. Find the nth partial sum of the Telescopic series and verify if its

convergent

(A telescopic series is a series whose terms can be rewritten so that most of them cancel out)

c.

Find the nth partial sum of the following series

1

1

n

and verify if its convergent

d.

Find the nth partial sum of the following series: 1, -1, 1, -1, 1 … and verify if its

convergent

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Example:

(Optional)

Find the nth partial sum of the Telescopic series

1

(

1

)

1

n

n

n

, and verify if it’s convergent

Geometric Series

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Operations with Series

If

1 n

n

a

is convergent,

1 n

n

b

is convergent, and c is a constant then the following series are also

convergent and:

1-

   

1 1

.

n n n

n

c

a

a

c

2-

     

1 1 1

)

(

n n n n n n

n

b

a

b

a

3-

     

1 1 1

)

(

n n n n n n

n

b

a

b

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Example: Find the sum of the series

  

1 1

)

5

2

4

3

(

k k k

Exercises:

Determine whether the series converges, and if so find its sum

1-

0

4

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Test for Divergence.

The converse is not true in general.

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Example

Show that

1

sin

n

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THE INTEGRAL TEST

Suppose f is a continuous, positive, decreasing function on [1,∞), and let a

n

= f(n). Then the

convergence or divergence of the series

1 n

n

a is the same as that of the integral

1

) (x dx f

i.e.:

(1) If

1

) (x dx

f

is convergent then

1 n

n

a

is convergent.

(2) If

1

) (x dx

f

is divergent then

1 n

n

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Example

Use the integral test to prove that the harmonic series diverges.

The p-series

The following series is called p-series where p is a positive number:

 1

1

n p

n

Prove:

Optional

Example

Determine whether the series converges, and if so find its sum

....

...

4

1

3

1

2

1

1

3 3

3

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COMPARISON TEST

Which test do you use for the following:

Suppose that

a

n

and

b

n

are series with positive terms and suppose that a

n

˂ b

n

for all n. Then

(1) If

b

n

is convergent then

a

n

is convergent.

(2) If

a

n

is divergent then

b

n

is divergent.

 1

2

2

1

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THE LIMIT COMPARISON TEST

Suppose that

a

n

and

b

n

are series with positive terms; if

c

b

a

n n n

lim

, where c is a finite

strictly positive number, then either both series converge or both diverge.

Example:

Determine whether the series

1

1

4

2

1

n

n

converges or diverges.

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THE ALTERNATING SERIES TEST

If the sequence of positive terms b

n

verifies

(1) b

n

is decreasing.

(2)

lim 0

  n n b

then the alternating series

1

1

)

1

(

n

n n

b

= b1 − b2 + b3 − b4 + · · · converges.

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Absolute Convergence

A series

1 n

n

a

is called absolutely convergent if the series of absolute values

1 n

n

a

converges.

-

Absolute convergence → convergence

-

The converse is not true in general.

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THE RATIO TEST

(1) If

n n n

a

a

1

lim

 

= L < 1 then the series

1 n

n

a

is absolutely convergent.

(2) If

n n n

a

a

1

lim

 

= L > 1 (including ∞) then the series

1 n

n

a

is divergent.

(3) If

n n n

a

a

1

lim

 

= 1 then the test is inconclusive (we do not know whether the series converges or

diverges).

Example:

Test the series for absolute convergence using the ratio test:

1-

1

!

1

n

n

2-

1

2

n

n

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3-

1

!

)

1

(

n

n n

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A

POWER SERIES

The interval of convergence is the set of values of

x

for which the series converges.

Example:

The following series converges to the function shown for

1

< x <

1:

...

...

...

...

...

0

 

n n

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Convergence of Power Series:

For a given power series

0

)

(

n

n n

x

a

c

there are only three

possibilities:

(1) The series converges only for

x

=

a

.

(2) The series converges for all

x

.

(3) There is a number

R

, called radius of convergence, such that the series converges if

|x−a| <

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Examples:

Find the radius of convergence and interval of convergence of the series

 

0

)

3

(

n n

n

x

Examples:

Find the radius of convergence and interval of convergence of the series

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Maclaurian Series:

It is a power series of the form

0 1 2 2

...

....

0

  n n n n

n

x

c

c

x

c

x

c

x

c

where

x

is a variable of indeterminate. It can be interpreted as an infinite polynomial.

Taylor Series

Given a fix number

a

, a power series in (

x − a

), or centered in

a

, or about

a

, is a series of the

form:

....

)

(

...

)

(

)

(

)

(

0 1 2 2

0

  n n n n

n

x

a

c

c

x

a

c

x

a

c

x

a

c

Representation of Functions as Power Series

The sum of the series is a function:

 

0

)

(

n n n

x

c

x

f

or

 

0

)

(

n n n

x

a

c

x

f

Two conditions:

1.

A value of x lying in the domain of the function must be chosen for the expansion point, a;

2.

Second, the function must be infinitely differentiable at the chosen point in its domain.

Taylor Series

Given a fix number

a

, a power series in (

x − a

), or centered in

a

, or about

a

, is a series of the

form:

....

)

(

...

)

(

)

(

)

(

0 1 2 2

0

  n n n n

n

x

a

c

c

x

a

c

x

a

c

x

a

c

....

)

(

!

)

(

...

)

(

!

2

)

(

)

(

!

1

)

(

'

)

(

)

(

) ( 2 '' 0

  n n n n

n

x

a

n

a

f

a

x

a

f

a

x

a

f

a

f

a

x

c

The Maclaurin Series: Taylor series with a=0

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Example:

Suppose the requirements of finding a power series are satisfied, find the Maclaurin Series

Expansion for

1- e

-x

2-

x

1

1

3-

x

1

1

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Operation, Differentiation and Integration of Power Series

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Example:

Suppose the requirements of finding a power series are satisfied, find the Maclaurin Series

Expansion for

1-

2

1

1

x

(39)

3-

2

)

1

(

1

x

Figure

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Referencias

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Related subjects : Sequences and series Power Series