Oct 23 12017 - University of Pittsburghkaveh/Lecture-MATH0230-Oct23-2017.pdf · Basic Properties of...
Transcript of Oct 23 12017 - University of Pittsburghkaveh/Lecture-MATH0230-Oct23-2017.pdf · Basic Properties of...
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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
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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 .
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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 , . . . .
.→?