c# pdf free : Add hyperlinks to pdf application SDK tool html winforms azure online TRENCH_REAL_ANALYSIS21-part239

202 Chapter4
InfiniteSequencesandSeries
Theseries
P
1
nD0
r
n
iscalledthegeometricserieswithratior.Itoccursinmanyappli-
cations.
Aninfiniteseriescanbeviewedasageneralizationofafinitesum
AD
XN
nDk
a
n
Da
k
Ca
kC1
CCa
N
bythinkingofthefinitesequencefa
k
;a
kC1
;:::;a
N
gasbeingextendedtoaninfinitese-
quencefa
n
g1
k
witha
n
D0forn>N.Thenthepartialsumsof
P
1
nDk
a
n
are
A
n
Da
k
Ca
kC1
CCa
n
; kn<N;
and
A
n
DA; nNI
thatis,thetermsoffA
n
g
1
k
equalthefinitesumAfornk.Therefore,lim
n!1
A
n
DA.
ThenexttwotheoremscanbeprovedbyapplyingTheorems4.1.2and4.1.8tothepartial
sumsoftheseriesinquestion(Exercises4.3.1and4.3.2).
Theorem4.3.2
Thesumofaconvergentseriesisunique:
Theorem4.3.3
Let
1
X
nDk
a
n
DA and
1
X
nDk
b
n
DB;
whereAandBarefinite:Then
1
X
nDk
.ca
n
/DcA
ifcisaconstant;
X1
nDk
.a
n
Cb
n
/DACB;
and
X1
nDk
.a
n
b
n
/DAB:
TheserelationsalsoholdifoneorbothofAandBisinfinite,providedthattherightsides
arenotindeterminate:
Droppingfinitelymanytermsfromaseriesdoesnotalterconvergenceordivergence,
althoughitdoeschangethesumofaconvergentseriesifthetermsdroppedhaveanonzero
sum.Forexample,supposethatwedropthefirstktermsofaseries
P
1
nD0
a
n
,andconsider
thenewseries
P
1
nDk
a
n
.Denotethepartialsumsofthetwoseriesby
A
n
Da
0
Ca
1
CCa
n
; n0;
and
A
0
n
Da
k
Ca
kC1
CCa
n
; nk:
Add hyperlinks to pdf - insert, remove PDF links in C#.net, ASP.NET, MVC, Ajax, WinForms, WPF
Free C# example code is offered for users to edit PDF document hyperlink (url), like inserting and deleting
pdf link to email; clickable links in pdf
Add hyperlinks to pdf - VB.NET PDF url edit library: insert, remove PDF links in vb.net, ASP.NET, MVC, Ajax, WinForms, WPF
Help to Insert a Hyperlink to Specified PDF Document Page
add links in pdf; add link to pdf file
Section4.3
InfiniteSeriesofConstants
203
Since
A
n
D.a
0
Ca
1
CCa
k1
/CA
0
n
; nk;
itfollowsthatADlim
n!1
A
n
exists(intheextendedreals)ifandonlyifA
0
Dlim
n!1
A
0
n
does,andinthiscase
AD.a
0
Ca
1
CCa
k1
/CA
0
:
Animportantprinciplefollowsfromthis.
Lemma4.3.4
Supposethatfornsufficientlylarge.thatis;fornsomeintegerN/
thetermsof
P
1
nDk
a
n
satisfysomeconditionthatimpliesconvergenceofaninfiniteseries:
Then
P
1
nDk
a
n
converges:Similarly,supposethatfornsufficientlylargetheterms
P
1
nDk
a
n
satisfysomeconditionthatimpliesdivergenceofaninfiniteseries:Then
P
1
nDk
a
n
diverges:
Example4.3.2
Considerthealternatingseriestest,whichwewillestablishlaterasa
specialcaseofamoregeneraltest:
Theseries
P
1
k
a
n
convergesif.1/
n
a
n
>0;ja
nC1
j<ja
n
j;andlim
n!1
a
n
D0:
Thetermsof
X1
nD1
16C.2/n
n2n
donotsatisfytheseconditionsforalln1,buttheydosatisfythemforsufficientlylarge
n.Hence,theseriesconverges,byLemma4.3.4.
Wewillsoongiveseveralconditionsconcerningconvergenceofaseries
P
1
nDk
a
n
with
nonnegativeterms. AccordingtoLemma4.3.4,theseresultsapplytoseriesthathaveat
mostfinitelymanynegativeterms,aslongasa
n
isnonnegativeandsatisfiestheconditions
fornsufficientlylarge.
Whenweareinterestedonlyinwhether
P
1
nDk
a
n
convergesordivergesandnotinits
sum, wewillsimplysay“
P
a
n
converges”or“
P
a
n
diverges.” Lemma4.3.4justifies
thisconvention,subjecttotheunderstandingthat
P
a
n
standsfor
P
1
nDk
a
n
,wherekisan
integersuchthata
n
isdefinedfornk.(Forexample,
X
1
.n6/2
standsfor
X1
nDk
1
.n6/2
;
wherek7.)Wewrite
P
a
n
D1.1/if
P
a
n
divergesto1.1/. Finally,letus
agreethat
X1
nDk
a
n
and
X1
nDkj
a
nCj
(whereweobtainthesecondexpressionbyshiftingtheindexinthefirst)bothrepresentthe
sameseries.
C# PDF Convert to HTML SDK: Convert PDF to html files in C#.net
Embed PDF hyperlinks to HTML links. How to Use C#.NET Demo Code to Convert PDF Document to HTML5 Files in C#.NET Class. Add necessary references:
add email link to pdf; add links to pdf online
VB.NET PDF Convert to HTML SDK: Convert PDF to html files in vb.
Turn PDF images to HTML images in VB.NET. Embed PDF hyperlinks to HTML links in VB.NET. Convert PDF to HTML in VB.NET Demo Code. Add necessary references:
pdf link open in new window; add url to pdf
204 Chapter4
InfiniteSequencesandSeries
Cauchy’sConvergenceCriterionforSeries
TheCauchyconvergencecriterionforsequences(Theorem4.1.13)yieldsausefulcriterion
forconvergenceofseries.
Theorem4.3.5(Cauchy’sConvergenceCriterionforSeries)
Aseries
P
a
n
convergesifandonlyifforevery>0thereisanintegerNsuchthat
ja
n
Ca
nC1
CCa
m
j< if mnN:
(4.3.3)
Proof
IntermsofthepartialsumsfA
n
gof
P
a
n
,
a
n
Ca
nC1
CCa
m
DA
m
A
n1
:
Therefore,(4.3.3)canbewrittenas
jA
m
A
n1
j< if mnN:
Since
P
a
n
convergesifandonlyiffA
n
gconverges,Theorem4.1.13impliestheconclu-
sion.
Intuitively,Theorem4.3.5meansthat
P
a
n
convergesifandonlyifarbitrarilylongsums
a
n
Ca
nC1
CCa
m
; mn;
canbemadeassmallaswepleasebypickingnlargeenough.
Example4.3.3
Considerthegeometricseries
P
r
n
ofExample4.3.1.Ifjrj1,then
frngdoesnotconvergetozero. Therefore
P
rdiverges,aswesawinExample4.3.1.If
jrj<1andmn,then
jA
m
A
n
jDjr
nC1
Cr
nC2
CCr
m
j
jrj
nC1
.1CjrjCCjrj
mn1
/
Djrj
nC1
1jrj
mn
1jrj
<
jrj
nC1
1jrj
:
(4.3.4)
If>0,chooseN sothat
jrj
NC1
1jrj
<:
Then(4.3.4)impliesthat
jA
m
A
n
j< if mnN:
NowTheorem4.3.5impliesthat
P
r
n
convergesifjrj<1,asinExample4.3.1.
LettingmDnin(4.3.3)yieldsthefollowingimportantcorollaryofTheorem4.3.5.
Corollary4.3.6
If
P
a
n
converges;thenlim
n!1
a
n
D0:
VB.NET PDF Page Replace Library: replace PDF pages in C#.net, ASP.
all PDF page contents in VB.NET, including text, image, hyperlinks, etc. Replace a Page (in a PDFDocument Object) by a PDF Page Object. Add necessary references:
check links in pdf; add links to pdf in acrobat
VB.NET PDF Thumbnail Create SDK: Draw thumbnail images for PDF in
PDF document is an easy work and gives quick access to PDF page and file, or even hyperlinks. How to VB.NET: Create Thumbnail for PDF. Add necessary references:
pdf edit hyperlink; add links to pdf file
Section4.3
InfiniteSeriesofConstants
205
ItmustbeemphasizedthatCorollary4.3.6givesanecessaryconditionforconvergence;
thatis,
P
a
n
cannotconvergeunlesslim
n!1
a
n
D0.Theconditionisnotsufficient;
P
a
n
maydivergeeveniflim
n!1
a
n
D0.Wewillseeexamplesbelow.
WeleavetheproofofthefollowingcorollaryofTheorem4.3.5toyou(Exercise4.3.5).
Corollary4.3.7
If
P
a
n
converges;thenforeach >0thereisanintegerKsuch
that
ˇ
ˇ
ˇ
ˇ
ˇ
X1
nDk
a
n
ˇ
ˇ
ˇ
ˇ
ˇ
< if kKI
thatis;
lim
k!1
X1
nDk
a
n
D0:
Example4.3.4
Ifjrj<1,then
ˇ
ˇ
ˇ
ˇ
ˇ
X1
nDk
r
n
ˇ
ˇ
ˇ
ˇ
ˇ
D
ˇ
ˇ
ˇ
ˇ
ˇ
r
k
X1
nDk
r
nk
ˇ
ˇ
ˇ
ˇ
ˇ
D
ˇ
ˇ
ˇ
ˇ
ˇ
r
k
X1
nD0
r
n
ˇ
ˇ
ˇ
ˇ
ˇ
D
jrj
k
1r
:
Therefore,if
jrj
K
1r
<;
then
ˇ
ˇ
ˇ
ˇ
ˇ
X1
nDk
r
n
ˇ
ˇ
ˇ
ˇ
ˇ
< if kK;
whichimpliesthatlim
k!1
P
1
nDk
r
n
D0.
Series ofNonnegativeTerms
Thetheoryofseries
P
a
n
withtermsthatarenonnegativeforsufficientlylargenissimpler
thanthegeneraltheory,sincesuchaserieseitherconvergestoafinitelimitordivergesto
1,asthenexttheoremshows.
Theorem4.3.8
Ifa
n
 0forn n  k;then
P
a
n
converges ifitspartialsumsare
bounded;ordivergesto1iftheyarenot:Thesearetheonlypossibilitiesand;ineither
case;
X1
nDk
a
n
D sup
˚
A
n
ˇ
ˇ
nk
;
where
A
n
Da
k
Ca
kC1
CCa
n
; nk:
.NET PDF SDK | Read & Processing PDF files
by this .NET Imaging PDF Reader Add-on. Include extraction of text, hyperlinks, bookmarks and metadata; Annotate and redact in PDF documents; Fully support all
add links pdf document; adding an email link to a pdf
PDF Image Viewer| What is PDF
advanced capabilities, such as text extraction, hyperlinks, bookmarks and Note: PDF processing and conversion is excluded in NET Imaging SDK, you may add it on
accessible links in pdf; add links to pdf
206 Chapter4
InfiniteSequencesandSeries
Proof
SinceA
n
DA
n1
Ca
n
anda
n
0.nk/,thesequencefA
n
gisnondecreasing,
sotheconclusionfollowsfromTheorem4.1.6
(a)
andDefinition4.3.1.
Ifa
n
0forsufficientlylargen,wewillwrite
P
a
n
<1if
P
a
n
converges.Thiscon-
ventionisbasedonTheorem4.3.8,whichsaysthatsuchaseriesdivergesonlyif
P
a
n
D
1. Theconventiondoesnotapplytoserieswithinfinitelymanynegativeterms,because
suchseriesmaydivergewithoutdivergingto1;forexample,theseries
P
1
nD0
.1/
n
os-
cillates,sinceitspartialsumsarealternately1and0.
Theorem4.3.9(TheComparisonTest)
Supposethat
0a
n
b
n
; nk:
(4.3.5)
Then
(a)
P
a
n
<1if
P
b
n
<1:
(b)
P
b
n
D1if
P
a
n
D1:
Proof (a)
If
A
n
Da
k
Ca
kC1
CCa
n
and B
n
Db
k
Cb
kC1
CCb
n
; nk;
then,from(4.3.5),
A
n
B
n
:
(4.3.6)
NowweuseTheorem4.3.8.If
P
b
n
<1,thenfB
n
gisboundedaboveand(4.3.6)implies
thatfA
n
gisalso;therefore,
P
a
n
<1. Ontheotherhand,if
P
a
n
D1,thenfA
n
gis
unboundedaboveand(4.3.6)impliesthatfB
n
gisalso;therefore,
P
b
n
D 1.
Weleaveittoyoutoshowthat
(a)
implies
(b)
.
Example4.3.5
Since
r
n
n
<r
n
; n1;
and
P
r
n
<1if0<r<1,theseries
P
r
n
=nconvergesif0<r<1,bythecomparison
test. Comparingthesetwoseriesisinconclusiveifr>1,sinceitdoesnothelptoknow
thatthetermsof
P
r
n
=naresmallerthanthoseofthedivergentseries
P
r
n
.Ifr<0,the
comparisontestdoesnotapply,sincetheseriesthenhaveinfinitelymanynegativeterms.
Example4.3.6
Since
r
n
<nr
n
and
P
r
n
D1ifr1,thecomparisontestimpliesthat
P
nr
n
D1ifr1.Compar-
ingthesetwoseriesisinconclusiveif0<r<1,sinceitdoesnothelptoknowthatthe
termsof
P
nr
n
arelargerthanthoseoftheconvergentseries
P
r
n
.
Section4.3
InfiniteSeriesofConstants
207
Thecomparisontestisusefulifwehaveacollectionofserieswithnonnegativeterms
andknownconvergenceproperties. Wewillnowusethecomparisontesttobuildsucha
collection.
Theorem4.3.10(TheIntegralTest)
Let
c
n
Df.n/; nk;
(4.3.7)
wheref ispositive;nonincreasing;andlocallyintegrableonŒk;1/:Then
X
c
n
<1
(4.3.8)
ifandonlyif
Z
1
k
f.x/dx<1:
(4.3.9)
Proof
Wefirstobservethat(4.3.9)holdsifandonlyif
X1
nDk
Z
nC1
n
f.x/dx<1
(4.3.10)
(Exercise4.3.9),soitisenoughtoshowthat(4.3.8)holdsifandonlyif(4.3.10)does.From
(4.3.7)andtheassumptionthatf isnonincreasing,
c
nC1
Df.nC1/f.x/f.n/Dc
n
; nxnC1; nk:
Therefore,
c
nC1
D
Z
nC1
n
c
nC1
dx
Z
nC1
n
f.x/dx
Z
nC1
n
c
n
dxDc
n
; nk
(Theorem3.3.4). FromthefirstinequalityandTheorem 4.3.9
(a)
witha
n
D c
nC1
and
b
n
D
R
nC1
n
f.x/dx,(4.3.10)impliesthat
P
c
nC1
< 1,whichisequivalentto(4.3.8).
FromthesecondinequalityandTheorem4.3.9
(a)
witha
n
D
R
nC1
n
f.x/dxandb
n
Dc
n
,
(4.3.8)implies(4.3.10).
Example4.3.7
Theintegraltestimpliesthattheseries
X
1
np
;
X
1
n.logn/p
; and
X
1
nlognŒlog.logn/
p
convergeifp>1anddivergeif0<p1,becausethesameistrueoftheintegrals
Z
1
a
dx
xp
;
Z
1
a
dx
x.logx/p
; and
Z
1
a
dx
xlogxŒlog.logx/
p
ifaissufficientlylarge. (SeeExample3.4.3andExercise3.4.10.) Thethreeseriesdi-
vergeifp 0:thefirstbyCorollary4.3.6,thesecondbycomparisonwiththedivergent
series
P
1=n,andthethirdbycomparisonwiththedivergentseries
P
1=.nlogn/. (The
208 Chapter4
InfiniteSequencesandSeries
divergenceofthelasttwoseriesforp  0alsofollowsfromtheintegraltest,butthe
divergenceofthefirstdoesnot.Whynot?)Theseresultscanbegeneralized:If
L
0
.x/Dx and L
k
.x/DlogŒL
k1
.x/; k1;
then
X
1
L
0
.n/L
1
.n/L
k
.n/ŒL
kC1
.n/p
convergesifandonlyifp>1(Exercise4.3.11).
Thisexampleprovidesaninfinitefamilyofserieswithknownconvergenceproperties
thatcanbeusedasstandardsforthecomparisontest.
ExceptfortheseriesofExample4.3.7,theintegraltestisoflimitedpracticalvalue,
sinceconvergenceordivergenceofmostoftheseriestowhichitcanbeappliedcanbe
determinedbysimplerteststhatdonotrequireintegration. However,themethodusedto
provetheintegraltestisoftenusefulforestimatingtherateofconvergenceordivergence
ofaseries.ThisideaisdevelopedinExercises4.3.13and4.3.14.
Example4.3.8
Theseries
1
X
1
.nCn/q
(4.3.11)
convergesifq>1=2,bycomparisonwiththeconvergentseries
P
1=n
2q
,since
1
.n2Cn/q
<
1
n2q
; n1:
Thiscomparisonisinconclusiveifq1=2,sincethen
X
1
n2q
D1;
anditdoesnothelptoknowthatthetermsof(4.3.11)aresmallerthanthoseofadivergent
series. However,wecanusethecomparisontesthere,afteralittletrickery. Weobserve
that
X1
nDk1
1
.nC1/2q
D
X1
nDk
1
n2q
D1; q1=2;
and
1
.nC1/2q
<
1
.n2Cn/q
:
Therefore,thecomparisontestimpliesthat
X
1
.nCn/q
D1; q1=2:
Section4.3
InfiniteSeriesofConstants
209
Thenexttheoremisoftenapplicablewheretheintegraltestisnot.Itdoesnotrequirethe
kindoftrickerythatweusedinExample4.3.8.
Theorem4.3.11
Supposethata
n
0andb
n
>0fornk:Then
(a)
X
a
n
<1
if
X
b
n
<1
and
lim
n!1
a
n
=b
n
<1:
(b)
X
a
n
D1
if
X
b
n
D1
and
lim
n!1
a
n
=b
n
>0:
Proof (a)
If
lim
n!1
a
n
=b
n
<1,thenfa
n
=b
n
gisbounded,sothereisaconstantM
andanintegerksuchthat
a
n
Mb
n
; nk:
Since
P
b
n
<1,Theorem4.3.3impliesthat
P
.Mb
n
/<1. Now
P
a
n
<1,bythe
comparisontest.
(b)
Iflim
n!1
a
n
=b
n
>0,thereisaconstantmandanintegerksuchthat
a
n
mb
n
; nk:
Since
P
b
n
D1,Theorem4.3.3impliesthat
P
.mb
n
/D1. Now
P
a
n
D1,bythe
comparisontest.
Example4.3.9
Let
X
b
n
D
X
1
npCq
and
X
a
n
D
X
2Csinn=6
.nC1/p.n1/q
:
Then
a
n
b
n
D
2Csinn=6
.1C1=n/p.11=n/q
;
so
lim
n!1
a
n
b
n
D3 and
lim
n!1
a
n
b
n
D1:
Since
P
b
n
<1ifandonlyifpCq>1,thesameistrueof
P
a
n
,byTheorem4.3.11.
ThefollowingcorollaryofTheorem4.3.11isoftenuseful,althoughitdoesnotapplyto
theseriesofExample4.3.9.
Corollary4.3.12
Supposethata
n
0andb
n
>0fornk;and
lim
n!1
a
n
b
n
DL;
where0<L<1:Then
P
a
n
and
P
b
n
convergeordivergetogether:
210 Chapter4
InfiniteSequencesandSeries
Example4.3.10
Withthiscorollarywecanavoidthekindoftrickeryusedinthe
secondpartofExample4.3.8,since
lim
n!1
1
.nCn/q
1
n2q
D lim
n!1
1
.1C1=n/q
D1;
so
X
1
.nCn/q
and
X
1
n2q
convergeordivergetogether.
TheRatioTest
Itissometimespossibletodeterminewhetheraserieswithpositivetermsconvergesby
comparingtheratiosofsuccessivetermswiththecorrespondingratiosofaseriesknownto
convergeordiverge.
Theorem4.3.13
Supposethata
n
>0;b
n
>0;and
a
nC1
a
n
b
nC1
b
n
:
(4.3.12)
Then
(a)
P
a
n
<1if
P
b
n
<1:
(b)
P
b
n
D1if
P
a
n
D1:
Proof
Rewriting(4.3.12)as
a
nC1
b
nC1
a
n
b
n
;
weseethatfa
n
=b
n
gisnonincreasing.Therefore,
lim
n!1
a
n
=b
n
<1,andTheorem4.3.11
(a)
implies
(a)
.
Toprove
(b)
,supposethat
P
a
n
D 1. Sincefa
n
=b
n
gisnonincreasing,thereisa
numbersuchthatb
n
 a
n
forlargen. Since
P
.a
n
/ D 1if
P
a
n
D 1,Theo-
rem4.3.9
(b)
(witha
n
replacedbya
n
)impliesthat
P
b
n
D1.
Wewillusethistheoremtoobtaintwootherwidelyapplicabletests:theratiotestand
Raabe’stest.
Theorem4.3.14(TheRatioTest)
Supposethata
n
>0fornk:Then
(a)
P
a
n
<1if
lim
n!1
a
nC1
=a
n
<1:
(b)
P
a
n
D1iflim
n!1
a
nC1
=a
n
>1:
If
lim
n!1
a
nC1
a
n
1
lim
n!1
a
nC1
a
n
;
(4.3.13)
thenthetestisinconclusiveIthatis;
P
a
n
mayconvergeordiverge:
Section4.3
InfiniteSeriesofConstants
211
Proof (a)
If
lim
n!1
a
nC1
a
n
<1;
thereisanumberrsuchthat0<r<1and
a
nC1
a
n
<r
fornsufficientlylarge.Thiscanberewrittenas
a
nC1
a
n
<
r
nC1
rn
:
Since
P
r
n
<1,Theorem4.3.13
(a)
withb
n
Dr
n
impliesthat
P
a
n
<1.
(b)
If
lim
n!1
a
nC1
a
n
>1;
thereisanumberrsuchthatr>1and
a
nC1
a
n
>r
fornsufficientlylarge.Thiscanberewrittenas
a
nC1
a
n
>
r
nC1
rn
:
Since
P
r
n
D1,Theorem4.3.13
(b)
witha
n
Dr
n
impliesthat
P
b
n
D1.
Toseethatnoconclusioncanbedrawnif(4.3.13)holds,consider
X
a
n
D
X
1
np
:
Thisseriesconvergesifp>1ordivergesifp1;however,
lim
n!1
a
nC1
a
n
D lim
n!1
a
nC1
a
n
D1
foreveryp.
Example4.3.11
If
X
a
n
D
X
2Csin
n
2
r
n
;
then
a
nC1
a
n
Dr
2Csin
.nC1/
2
2Csin
n
2
whichassumesthevalues3r=2,2r=3,r=2,and2r,eachinfinitelymanytimes;hence,
lim
n!1
a
nC1
a
n
D2r
and
lim
n!1
a
nC1
a
n
D
r
2
:
Therefore,
P
a
n
convergesif0 < r < 1=2anddivergesifr > > 2. Theratiotestis
inconclusiveif1=2r2.
Documents you may be interested
Documents you may be interested