MAT 1236Calculus III

Section 11.1

Sequences Part II

http://myhome.spu.edu/lauw

HW and …

WebAssign 11.1 Part II(10 problems, 26* min.)

From this section on, HW due at 1:45pm Lab 02 tomorrow Exam 2 is coming up...

Definition

A sequence is a collection of numbers with an order

, , , 321 aaa

Notation:

na 1nnaor

Definition

A sequence is convergent if

number finitelim n

na

Otherwise, is divergent

Limit LawsIf , are 2 convergent sequences and c is a constant, then

0 and 0 if limlim

lim

limlim

0lim if lim/lim)/(lim

limlim)(lim

limlim)(lim

np

nn

pn

n

n

nn

nn

nn

nn

nn

nnn

nn

nn

nnn

nn

nn

nnn

apaa

cc

acac

bbaba

baba

baba

Finding limits

There are 5 tools that you can use to find limit of sequences

Tool #1 (Theorem)

If nanf )( Lxfx

)(lim , then Lann

lim and

Tool #1 (Theorem)

If nanf )( Lxfx

)(lim , then Lann

lim

.

and

naxf ),(

nx,

L

1

)1(1 fa

)2(2 fa

)(nfan

2 n

Tool #2

Use the Limit Laws

Tool #3 (Squeeze Theorem)

If and

Then

Lca nn

nn

limlim

Lbnn

lim

0for nncba nnn

Tool #3 (Squeeze Theorem)

If and

Then

Lca nn

nn

limlim

Lbnn

lim

.

nan

L nb

nc

0for nncba nnn

Example 4

n

n

n

)1(2lim

Example 4

n

n

n

)1(2lim

2 ( 1)

2 ( 1)

n

n

n

n

Remark

0)1(2

lim

theorem,squeeze By the

n

n

n

It is important to state

the name of the theorem

Tool #4 (Theorem)

0lim then ,0lim If n

nn

naa

Tool #4 (Theorem)

0lim then ,0lim If n

nn

naa

.

na

n0

na

Tool #4 (Theorem)

0lim then ,0lim If n

nn

naa

.

na

n0

na

Example 5

n

n

n

)1(lim

PPPFTNE: Correct Answer?Correct Argument?

( 1) 1 1lim lim( 1) lim( 1) lim

lim( 1) 0

0

nn n

n n n n

n

n

n n n

Remark

0lim then ,0lim If n

nn

naa

.

na

n0

na

zerowithonlyworks

Remark

If lim 1, then lim ?n nn n

a a

.

na

n1

na

zerowithonlyworks

0

1

Tool #5

0 if 1 1

lim 1 if 1

(D.N.E.) otherwise

n

n

r

r r

1 ror

Example 6

n

n

n 3

2lim

1

0 if 1 1

lim 1 if 1

(D.N.E.) otherwise

n

n

r

r r

Expectations

Must give the precise criterion -

21

3

Hint on HW #10

Use the “cruise ship” theorem Use l'Hospital's rule Be very careful on your arguments and

presentation Solution will be posted (?) tomorrow

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