Math Analysis I - Limits of Functions Part 2 (Sequences)

Sequences X - I N

A remarkable special case of function arises when and the domain contains a

{ h e n a n o }

l

I N

set of the type for a certain natural number n0 ≥0.

Such a function is called sequence. The image of the natural number n is denoted by

the symbol ; thus we shall write .

Q : r a n

h e >

r a n (Qu)

A common way to denote sequences is (ignoring possible terms with n < n0)

n i n o

led

or . N A I R

f :

Generally, a function is called a sequence.

It is convenient to think of a sequence as of a lots of images.

(fu) I t , Iz---tu}

Iz,

=

flu) - a n 2,3...}

{an}

{en} 10,1,

e-g. =

n e w

• ,

¥ 1 3 1 1 7

1 ^

H e

I - I o .

2

- . Oh

it, Nz

d o N z

2. Math Analysis Limits of Functions 1 of 65

It}uelN-lol

• 42

1/3 1

Yu,

°

1 ¥

02

oyez Q,

1 1 so

k e .

z -1/2 •

11=1/3 . .

ly

l y du-5

Q,

{sinn}neN

• f¥÷¥E÷¥.

L-Qi K

I - I - - - - t

-

L -

- o

{¥}neN

• 1 9 . 1 2 4 3 3 ¥ .

Ll 0203

do

I--=#=-x-

€ ±

2- x Oz %

Q, Oz Uh %

2. Math Analysis Limits of Functions 2 of 65

u

{ s t F)

• new-l" %6¥±

S i s

1- 256

ily@a

i l y Q,

1

2 f

- - - - - - - - - - - - - - - -

- x - ± ± = ± = =

-

1 1 ¥ 1 1 Qt

ez ez 95

Q, dh %

{h!}nEN

. 161---------x

¥÷÷÷±.

%

02 93 du 05 07

l ,

2. Math Analysis Limits of Functions 3 of 65

Notation

Real sequences can be defined as functions:

flu)

ran =

and can be defined by recurrence:

R

{do E flea)

E

tent, R

Leo E

e . g.. IR

- l - d u L E

tent, , ez-fez-13

1 2 ,

too le,

el, e z - d o

= =

duel" - e ,

6 . i)

{do E [0,4]

• (I-en),

e, - µ .eu µ -

2. Math Analysis Limits of Functions 4 of 65

Asymbtotic behaviour

The concern of the studies about sequences of the type is their

d u

asymptotic behaviour for an infinitely large n.

The aim is to identify what values the sequence will eventually assume.

¥ ,

(u) tu,

e-g.. e n - s

- a

u - s e n

i ¥ ¥

l'T}

• yo,

n e w though %,

a n O n e v e r

E e v e n

r e a c h e s a 0"

"air close

arbitrarely t o

is

(Yue

(Sin

• dense

Ou the

is in

µ E- s ]

L ,

s e t

I l

o

-

11111111111111141111111111111111111111 if

{-1

1 is

n even

{-t"} ten-

n e w odd

• if is

n

± ¥

The asymptotic behaviour of sequences is generally unpredictable.

However, there are some particular cases in which it can be easily described.

2. Math Analysis Limits of Functions 5 of 65

Converg

encelet b e real

a sequence,

o u said point

is t o

t o t h e

du converge

R

l is:

E

HE F r i e IN t h a t fo r 5,

s o , u s

such

l e u - e l E E t h a t

eventually-a i s :

,

(eu. f )

d the

where

E , dlau, e) is

e distance e n t o l

from

l

said t o and

to

is

a n converge

l said be l i m i t

t h e of

is to e n

a u # l l

lim o u

; s

n or

→

The set in which converges is better to be specified with the formula:

Q n

“ ”

X

convergent

Q u i n

Remarks: lien-el Ta n e t t e

t e # l - E

given

t e x t

2. Math Analysis Limits of Functions 6 of 65

htt""}u£

e . g .

- who}

1/2

t h

1/3 O

T I X

- 03

Q, e z

da

e n O

i → ¥1 is

n . e v e n

{an}neµ are

• odd

is

u

lol

\ I

n

2/3

K 4547

ke K

¥ * ¥

Gen 0 5 9

03

e z

Q, ¥ i x ¥ ¥

the different

behaviour

asymptotic is o n

sides

the t w o

2. Math Analysis Limits of Functions 7 of 65

Representation

l - E l l t e f o r

1 i n

it,

¥ 2

l-E, lte,

the trees

larger

the hold

t h e i t :

meeee e ,

s

I depends E

o n l - E

for T E l

eventually

w i l l be trapped:

au i a u t e

1,

t : #

2 €

-

↳

ProvefoilHnesient

let E> o

I / E

t

e n - 5

I I-O fir IN

E

T

e n - 5 E

I /

th t h ,

fixing e n - 5 E

it t

2. Math Analysis Limits of Functions 8 of 65

provethetsonlIITlneN.am#

arbitrary E

I t o

choose a n

fix

2- find i t t h a t

such e s u c h

i I en-st

tn i n , E

T

let chosen

arbitrary

be

E t o /

ten-s r e

'k¥4-she

I#i¥#lie

/#/ It,

⇐ r e

=

En'

↳ 3 t t E

3¥

my

↳ 3 - E

1 E 3 - a

i T O

e -

321 An

ni IN

y e Ien-el

thus, t u r n ,

fixing Te o , E E

3¥,

2- E e 3 - a o r tf,

Fire

3¥ IF

Fn

m y IN IN>

E

FIFTH, lie

t u r n # t e n - I

thus fixing

2. Math Analysis Limits of Functions 9 of 65

Provethetsorlff.fi/nElN,en#

ten-of

let r e

t h a t

s u c h

E t o

1231571=2+3%5 E 5N

5h' E - 3 h

1 3 2

i t

E 29

E

t t o

n 3t,o{T#U 3-251

r

u s o h e 10 E

a-13%11) [3%11]

v

f i x i n g t i

/ /

thin E

t

e u

3h

I E

=

given ¥ 2 12+3=15

35¥-free e

is

¥ "5¥

⇐

E "

e (E) lent

fixing E

turn,

i t - 5

I

t

2. Math Analysis Limits of Functions 10 of 65

pnouetheteu-siulutli.sc#geut

th

(hit) N

0

s i n E

= ( u t )

then, e n - s i n i s constant sequence

a

lenke

l e t E> o

d e f i n i t i o n

b y o f convergence, b u t 0

Provetheteuesincuttonvergent

IT

consider periodicity

t h e t h e function

s i n

o f

NII-Lo,

if z i t ] ,

t u n e i n t h e n

l in U I - I R

e a t 2 a}

{O,

N , 3,

A ,

since E L ,

n -

n

do l , = - 1 ;

e z - O e a t

1 ;

O ; ez o

;

=

= F

F

l o t l, ez

e z e ,

= (UI)}

{sin l

i f

therefore Not

CONVERGENT

2. Math Analysis Limits of Functions 11 of 65

Theorem (uniquenessoythmitt

The limit of a convergent sequence is unique.

Proof ( b y a n t i t i o n )

suppose: l, R

l R E

and

E →

a n

→

o u Il I

- l ,

L F

t h a t t o

l ,

s u c h e - ¥

choose 2 l IR,

E

→

r a n

1 since

Fa, Ien-l l

1N, e

r

E it,

u >

l, IR,

E

2- →

r a n

s i n c e

Fr, Ion-l,

1N, l E

nine e

E viz)

for ( i t , ,

T - m e x 5

a n d h t

ten-l,

I Lee

e n - l A hold

l a t

E t i m e

t h e

e s a m e

I - I l - e n

l l - l ,

then, l e le-enl t

t h e re ,

e n - l ,

t

leu-el

l e

le-e, t h e n - l i l

- a Il-l, e-le-l,

I l

E E E

t 2

→ =

Il l e l l - l i l

- l , ABSURD

2. Math Analysis Limits of Functions 12 of 65

Provethetenetitishotcouvergent

I i a

e 2kt 2 K

I (constant sequence) l e t

1 e n - t

h i s e v e n

te n - H e e H E

e n e

I - E x

take r e n t

O 2

E - 1 ( c o n s t a n t

odd seauence)

2- h i s let

e n - - 1

lentil o u t - I t

t E

e - l - q t

f Q u

2 E O

- {0}

A-2rem t o

o r u r 2

e - =

2. Math Analysis Limits of Functions 13 of 65

Determineiftheseguenceeniscouvergent

{ 'I m e "

" i s 1N-lo}

e n - n e

y a odd

i s

u Len l e E

it,

a n # o

1 is n >

u e v e n , ten-11

odd,

2- m i n ,

t r e

e n

i s ' →

n -121

I l ,

l e t ez

since t o

,

E - ¥

choose 2

(but l E-Ili-ld-t

t h e n - I E t E - 2

t te n - f u

I e n t i l

I t t e n - I l ' s absurd

1 Which

I 1 i s

2. Math Analysis Limits of Functions 14 of 65

(permanencies )

Theorem IR, l

l e

↳

o n s o

{on} e v e n t u a l l y

then, positive

is

F T IN

h e m e l y, t h a t

such i n # a n >

E n o

Proof let E-l>

1 0 /

FEE ten-L

IN i t E

t h a t r

n >

s u c h

⇐ l - e s e l

e n t e

⇐ 2L

d u e

o r

Remain: {eu)

i f eventually

then

l i o , i s

negative

let end

2- l e o - l

e - en-el

I

Fir IN a t e

t h a t n >

s u c h

E

⇐ l - E l

t e n t e

t

⇐ z e e e u r o

2. Math Analysis Limits of Functions 15 of 65

Corollary: e n - l l ± o

o u z o ,

Pros:

by contradiction, suppose: (by

L e o theorem)

eventually o n e 0 ABSURD

Remote: l±o

e u r o

I n true

particular that

not

i t is a

convergent has

p o s i t i v e a

s e q u e n c e

l i m i t

p o s i t i v e

strictly {I} e n >

e . g. 0

new-lo,

him e n - o

2. Math Analysis Limits of Functions 16 of 65

Theorem ( d o u b l e c o m p a r i s o u r s e u d w i "Faenanmen"

{Zu}

Given Lyn},

{Xu},

the sequences ' l

l IR

IR

t h a t :

such zu E

yur> E ,

yuk

if I

Xu Zu

then, l

Xu →

Proof

let arbitrary

E > o l,

since

1 Yu ' →

Fa, l t

IN l-Eryne E

I,

t h a t

E s u c h n >

zu-=L,

2- s i n c e

Fitz ft

IN l - E r E

I,

t h a t Zone

E s u c h n >

521

for ( i i i . t a k e i n

T - m a x n

l t

l - E e

then, E

Xue Zai

yue

by l

definition limit, Xu

of →

2. Math Analysis Limits of Functions 17 of 65

Pnouethetforlutftnc-N-l.is/dn#-

eveutueeeyuentlen2 55

i t

A

IN u t - u - l

since f o r

n e 7 0 n = - 2

fixing 2

us, ¥.

¥5#e ut

by sandwich, 0

' →

Remarks: Ilu) polynomial

if is n

i n

a

-IE, n"

teh'

Icu) n' t . . . te,

t e,

e ,

e, n

t

t 0

a

Ecu)

Pouuebyseudwicuthetle.tn/i#

IN

em,

e u e E

e v e n t u a l l y, I n

e

↳ et

F e e t ⇐ oeetnet

test

since ,

by h e

sandwich e - o -

2. Math Analysis Limits of Functions 18 of 65

Theorem ( t a r g e t e d )

A convergent sequence is bounded. Namely:

FM 112 leule M

that

s u c h

E

Proof htt

e- e

,

# l ,

since for E - L ,

an →

FT IN i t

t h a t

E n >

s u c h e l

l - e t e

e e n

l - I l

t i

e a t I

then, A

there s e t

finite

exist a er}

{l-yeti,

A

tuet

s u c h l o , e,,...

=

l Al I t 3

w i t h = Timex,<