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Lecture 20

# Math1xx3 Lecture 20: Lecture 20

Department
Mathematics
Course Code
MATH 1XX3
Professor
Lecture
20

Page:
of 3
Testsforconvergencestrategi
Recall:¥"
converges.What does it converge to ?
Let hn :1t
÷+'s +. . .
+'s
san =I-'s +5 -
...
+znt,
-In
Claim :San =han -hn Induction !
Basqfgse :so =I-st ='S =(
+
F) -I=hz -h,(holds)(want to show Sai =ha ;-hi
Induction :Assume true for all ich .
Show true for n
San =Sxn ",ttf -ant
=Lhacn
.
a
-
hn.lt K-ztn
.
=#n¥n
-
*-(hn -
NH )+than
=han .hntnztn+'T -anti +£.
=han -hn .
Sequence :tn =(1+ It ztt .. "th -bn (n))=hn -kln )has alimit 8
Proof :
.
ftx÷
trea under curve .
.fi's dx =tax],
"
iHoo
=..aurum
So bn :It ts + tz +
...
+th -hen)=
"
over area count
"
20
.
since :tn -th ,,
=fit 's +
...
+'T )-hen ))-(I+
.. . .+n÷ -bn (NHD
=-bn (n)th (hit )-
,
=+bn (F) 20
Point :th is
asequence ofpositive numbers where tnztnh for all n
So him.tn exists and call it 8
Claim :a¥''
=lnCD
hignosan =nkjmslhzn
-hn )=ntemdhn
-
M-than
-
H
=him ,
11+'s +
...
+th)-1It 's ttn -born )
)-(Htst ...int -tnln))
rims km .hail -him .tn#=hH
So :
,
%"
-
'h( 21 OR Is IT ?
We have :1-tzt 's -Tt
¥-5tzt -tg ....
=hk ).C
Multiply both
sides by 2:
T.tt *to -t+to -ist '
Ty -to ... =That ='s C
:-tz to tty to tot to .tg
...
=÷h( 2) =TC
.
+2¥
+}-tat 's +0 +'s .Ts ...3zh(21=22 C
.
Same terms in different order
What's Happening ?,
'
±n
"
converges conditionally
(Riemann )If ¥converges conditionally and ris areal number,then
there is arearrangement of Ean so that it converges to r
Strategies :
!1!If ofthe form §'F P.series
!2!If of the form TG
,
arm use geometric series
!3!If it is similar top -series or geometric series ,use comparison or limit .
comparison test
!4!If him.
an to ,use the divergence lest
!5!If it has form t
't "
bn ,use Alternate
Series Test
@If it has factorials or other products,use the ratio test
(careful : no conclusion if him ,19ham
=
l)
!7!If an has the form an =bn
"
,use the root test
!8!If an =fln)where f?Stx)dx can be evaluated easily,try
integral lest
(caution :be aware at hypothesis lfcx
)
has to be positive,continuous ,and decreasing))