### 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*

### }

*n*

_{}1

**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: **

### 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}

*n*

### 1

### lim

###

###

_{n}

_{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. }

_{ if b ≠ 0. }

**Example: Evaluate the limit, if it exists. **

### 1-3

### 9

### 6

### 3

### lim

_{2}2

###

###

###

_{n}

_{n}

*n*

*n*

*n*

### 2-3

### 5

### 9

### 6

### lim

_{3}2

###

###

###

_{n}

_{n}

*n*

*n*

*n*

### 3-3

### 9

### 6

### 2

### lim

2###

###

###

_{n}

_{n}

*n*

*n*

*n*

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

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

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

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

**Example: Using L'Hopital's Rule **

**Example: Using L'Hopital's Rule**

### Determine if the sequence

*n*

*n*

*a*

_{n}###

### ln

### is convergence or divergence

**Example **

**Example**

### Find

###

###

*n*

*n*
*n*

*n*

*a*

### lim

*r*

### lim

### 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.

**Example: **

**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*

**Bounded **

**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: **

**Example:**

### Prove that the sequence

*n*

*n*

*a*

_{n}###

###

### 1

* is bounded *

**Example: **

**Example:**

### Prove that the sequence

*n*

*n*

**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 **

**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*

*n*lim

### 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*

**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 **

**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 **

**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*

**Example: Find the sum of the series **

###

###

1 1### )

### 5

### 2

### 4

### 3

### (

*k*

*k*

*k*

**Exercises: **

**Exercises:**

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

*1-*

###

0

### 4

**Test for Divergence. **

### The converse is not true in general.

**Example **

### Show that

###

1

### sin

*n*

**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*

**Example **

**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**

**Example**

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

### ....

### ...

### 4

### 1

### 3

### 1

### 2

### 1

### 1

3 3

3

###

###

###

**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

**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: **

**Example:**

### Determine whether the series

###

1

### 1

###

### 4

2### 1

*n*

*n*

### converges or diverges.

**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

###

###

11

### )

### 1

### (

*n*

*n*
*n*

*b*

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

**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.

**THE RATIO TEST **

### (1) If

*n*

*n*

*n*

_{a}

_{a}

*a*

1
### lim

### = L < 1 then the series

###

1
*n*

*n*

*a*

### is absolutely convergent.

### (2) If

*n*

*n*

*n*

_{a}

_{a}

*a*

_{1}

### lim

### = L > 1 (including ∞) then the series

###

1
*n*

*n*

*a*

### is divergent.

### (3) If

*n*

*n*

*n*

_{a}

_{a}

*a*

_{1}

### lim

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

### diverges).

**Example: **

**Example:**

### Test the series for absolute convergence using the ratio test:

### 1-

###

1

### !

### 1

*n*

*n*

### 2-

###

1

### 2

*n*

*n*

### 3-

###

###

1### !

### )

### 1

### (

*n*

*n*
*n*

**A **

**POWER SERIES **

**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*

**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| < *

**Examples: **

**Examples:**

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

###

###

0### )

### 3

### (

*n*

*n*

*n*

*x*

**Examples: **

**Examples:**

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

**Maclaurian Series: **

**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 **

**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

**Operation, Differentiation and Integration of Power Series **

**Example: **

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

### Expansion for

### 1-

_{2}

### 1

### 1

*x*

###

### 3-

_{2}