Oct 23 12017 - University of Pittsburghkaveh/Lecture-MATH0230-Oct23-2017.pdf · Basic Properties of limit of sequences in {.} anttbn = himFlim{an] {bn} n → a × n→oo × n→a

Oct . 23 12017%

- @Review of topics so far

.nation

/substitution

•Methods of int - trig .

sub.

T int . by parts

•Int .

with infinite limits ) partial fraction

• volumes ( using integration ) & Are length .

• Applications in physics-

• Some 3D geometry-

• parametric Armes & polar Coon-Newtopic

Sequences & series

real

A"

sequence"

( of numbers ) :

a, saz , a } , . - - . ( repetition allowed )

Ex .1

,2

, 3.4 - - - -

Notation :

•

{ n } %, ± ,l 1

'n } - → { 9

.}n= ,

{ ⇒ /

{ ⇒ 1'

' 4- ' Fists . . .

{ 1 } 61

9 19 1g Is .

- . .

{ e,,n+j

AgDay1 , -1 , 1 , -1 ,

-. .

.

n=l n=2

e)2

←,)3

Imoprtant concept : Limitofaseque_( analogue of limit of a function y=f( × , ) .

means :

lim an =L as n - a

" → •

( amnion ,

gynsI.tn?odIeTCL is called

"

limit " of

the sequence { an }

.

Ex.

an=h lim n = + A

{ n }h→o

an = he1in b- =D

n→a

an = 'nz nlignantz=0

an =L lim I = Intl n→a

an =fD nil

£,

-1 , ) ,- ) , ...

lime ) does not exist .

n → a

Re=sedefoflimit : lim An =Lsmall n→A

For any Do( we would to get close to L

Large closer than E ,provided

there [email protected] enough )

such that if n > N then lan.LI#d

- dist . of an & L

is < E .

BasicProperties of limit of sequences

.in { anttbn} = him { an ]Flim { bn }n → a

× n→oo × n→a

• nljz{g÷} = k¥41 providedlim { bn } that

n→a him { bn } # °

p p n→a

.lim {9n} = dim{and

n→aon→a

@fixed )

•lim Ian ) =o # lim an =D

n→a÷• tim T =0 mns easy to show using

n→a definition .

want :

Take any e > 0 |n1→|< E

÷' < nE

Take N > at @

Then n > N > et ⇒ f- < e i. e .

tin 'n-=0n→oo

- ( infinite sum )

Series : sum of elements of a sequence .

{ an } - a,

+92+93-1 . . . .

a,

s qtaz , Astaztaz , 91+92+93+94 , . . . .

.→?