{ } { S n } is monotonically decreasing if Sn

Transcription

1 Sequece A sequece is fuctio whose domi of defiitio is the set of turl umers. Or it c lso e defied s ordered set. Nottio: A ifiite sequece is deoted s { } S or { S : N } or { S, S, S,...} or simply s { S } e.g. i) { } {,,3,... } ii) 3,,, ( ),,,,... iii) { } { } Susequece It is sequece whose terms re cotied i give sequece. A susequece of { } Chpter Sequeces d Series Suject: Rel Alysis Level: M.Sc. Source: Syed Gul Shh (Chirm, Deprtmet of Mthemtics, UoS Srgodh) Collected & Composed y: Atiq ur Rehm(mthcity@gmil.com), S is usully writte s { S }. Icresig Sequece A sequece { S } is sid to e icresig sequece if S S +. Decresig Sequece A sequece { S } is sid to e decresig sequece if S S +. Mootoic Sequece A sequece { S } is sid to e mootoic sequece if it is either icresig or decresig. { S } is mootoiclly icresig if S S + or S + S, S { S } is mootoiclly decresig if S S + or, S + Strictly Icresig or Decresig { S } is clled strictly icresig or decresig ccordig s S > + S or S < + S. Beroulli s Iequlity Let p R, p d p the for we hve ( p) + > + p : We shll use mthemticl iductio to prove this iequlity. If L.H.S ( + p) + p+ p R.H.S + p LHS.. > RHS..

2 i.e. coditio I of mthemticl iductio is stisfied. Suppose ( ) + Now ( + p) ( + p)( + p) ( p)( p) Sequeces d Series p > + p...() i where > + + usig (i) + p+ p+ p + ( + ) p+ p + ( + ) p igorig + ( ) ( ) p + p > + + p Sice the truth for implies the truth for + therefore coditio II of mthemticl iductio is stisfied. Hece we coclude tht ( p) + > + p. Let S + where To prove tht this sequece is icresig sequece, we use p, i Beroulli s iequlity to hve > + > + > + S > S S is icresig sequece. which shows tht { } + Let t + ; the the sequece is decresig sequece. We use p i Beroulli s iequlity. where + > +.. (i) (ii) Now t from (ii)

3 Sequeces d Series > + + > + + t + i.e. t > t Hece the give sequece is decresig sequece. from (i) > Bouded Sequece S is sid to e ouded if there exists positive rel umer λ A sequece { } such tht S < λ N If S d s re the supremum d ifimum of elemets formig the ouded sequece { S } we write S sup S d s if S All the elemets of the sequece S such tht S < λ N lie with i the strip { y: λ < y< λ}. But the elemets of the uouded sequece c ot e cotied i y strip of fiite width. s (i) { } U (ii) { V } { } ( ) is ouded sequece si x is lso ouded sequece. Its supremum is d ifimum is. (iii) The geometric sequece { } ouded elow y. π (iv) t is uouded sequece. r, r > is uouded ove sequece. It is Covergece of the Sequece A sequece { S } of rel umers is sid to coverget to limit s s, if for every positive rel umer ε >, however smll, there exists positive iteger, depedig upoε, such tht S s < ε >. Theorem A coverget sequece of rel umer hs oe d oly oe limit (i.e. Limit of the sequece is uique.) : Suppose { } S coverges to two limits s d t, where s t. Put ε s t the there exits two positive itegers d such tht S s < ε > d S t < ε > S s < ε d S t < ε hold simulteously > mx(, ). Thus for ll > mx(, ) we hve s t s S + S t

4 Sequeces d Series s t S s + S t < ε + ε ε s t < s t < s t Which is impossile, therefore the limit of the sequece is uique. Note: If { S } coverges to s the ll of its ifiite susequece coverge to s. Cuchy Sequece A sequece { x } of rel umer is sid to e Cuchy sequece if for give positive rel umer ε, positive iteger ( ) ε such tht x xm < ε m, > Theorem A Cuchy sequece of rel umers is ouded. Let { S } e Cuchy sequece. Te ε, the there exits positive itegers such tht S Sm < m, >. Fix m + the S S S + S + + S S + S + + < + S + > < λ >, d + ( chges s ε chges) λ S + Hece we coclude tht { S } is Cuchy sequece, which is ouded oe. Note: (i) Coverget sequece is ouded. (ii) The coverse of the ove theorem does ot hold. i.e. every ouded sequece is ot Cuchy. S where S ( ),. It is ouded sequece ecuse Cosider the sequece { } ( ) < But it is ot Cuchy sequece if it is the for ε we should e le to fid positive iteger such tht S Sm < for ll m, > But with m +, + whe + >, we rrive t S S ( ) ( ) m < is surd. Hece { S } is ot Cuchy sequece. Also this sequece is ot coverget sequece. (it is oscilltory sequece)

5 Sequeces d Series Diverget Sequece S is sid to e diverget if it is ot coverget or it is uouded. A { } e.g. { } is diverget, it is uouded. (ii) {( ) } teds to or - ccordig s is eve or odd. It oscilltes fiitely. (iii) {( ) } is diverget sequece. It oscilltes ifiitely. Note: If two susequece of sequece coverges to two differet limits the the sequece itself is diverget. Theorem If S < U < t d if oth the { S } d { } s, the the sequece { U } lso coverges to s. t coverge to sme limits s Sice the sequece { S } d { t } coverge to the sme limit s, therefore, for give ε > there exists two positive itegers, > such tht S s < ε > t s < ε > i.e. s ε < S < s+ ε > s ε < t < s+ ε > Sice we hve give S < U < t > s ε < S < U < t < s+ ε > mx(,, ) s ε < U < s+ ε > mx(,, ) i.e. U s < ε > mx(,, ) i.e. limu s Show tht lim Solutio Usig Beroulli s Iequlity Also < + lim lim < lim + lim < i.e. lim. ( ). > >..

6 Sequeces d Series Show tht lim ( ) ( ) ( ) + + Solutio We hve S ( + ) ( + ) ( ) d < S < ( ) < S < 4 lim < lims < lim 4 < lims < Theorem lims If the sequece { } S coverges to s the positive iteger such tht S > s. We fix ε s > positive iteger such tht S s < ε for > S s < s Now s s s s S s s+ S s s < S Theorem < ( ) Let d e fixed rel umers if { S } d { } respectively, the S + t coverges to s + t. (i) { } (ii) { } (iii) St coverges to st. S t t coverge to s d t coverges to s t, provided t d t. Sice { S } d { } t coverge to s d t respectively, S s < ε > N

7 t t < ε > N Also λ > such tht S < λ (i) We hve S + t s+ t S s + t t Sequeces d Series S > ( { } ( ) ( ) ( ) ( ) ( S s) + t ( t) This implies { S t } is ouded ) < ε + ε > mx(, ) Where ε ε + ε certi umer. ε (ii) (iii) + coverges to s + t. St st St St+ St st S ( t t) + t( S s) S ( t t) + t ( S s) < λε + t ε > mx(, ) where ε λε + t ε certi umer. ε This implies { } t t t t tt Theorem St coverges to st. t t ε t t t t < ε > mx(, ) ε 3 where 3 t This implies t coverges to t. S Hece S t t coverges to s s t t t t > t ε ε certi umer.. ( from (ii) ) For ech irrtiol umer x, there exists sequece { } umers such tht limr x. Sice x d x + re two differet rel umers rtiol umer r such tht x< r < x+ Similrly rtiol umer r r such tht x< r < mi r, x+ < x+ Cotiuig i this mer we hve x< r3 < mi r, x+ < x+ 3 x< r4 < mi r3, x+ < x x< r < mi r, x+ < x+ r of distict rtiol

8 Sequeces d Series This implies tht sequece { r } of the distict rtiol umer such tht Sice Therefore Theorem x < x< r < x+ lim x lim x+ x limr x Let sequece { } (i) If { } (ii) If { } S e ouded sequece. S is mootoiclly icresig the it coverges to its supremum. S is mootoiclly decresig the it coverges to its ifimum. Let S sup S d s if S Te ε > (i) Sice S sup S S such tht S ε < S Sice { S } is ( stds for mootoiclly icresig ) S S S S S ε < < < < + ε for > S ε < S < S + ε for > S S < ε for > lim S S (ii) Sice s if S S such tht S < s+ ε Sice { S } is. ( stds for mootoiclly decresig ) s ε < s< S < S < s+ ε for > s ε < S < s+ ε for > S s < ε for > Thus lims s Note : A mootoic sequece c ot oscillte ifiitely. Cosider { S } + As show erlier it is icresig sequece Te S + The S + S + S + Usig Beroulli s Iequlity we hve

9 Sequeces d Series S + > S <,,3,... S < 4,,3,... S < S < 4,,3,... S is ouded oe. Which show tht the sequece { } Hece { } + + S is coverget sequece the umer to which it coverges is its supremum, which is deoted y e d < e < 3. Recurrece Reltio A sequece is sid to e defied recursively or y recurrece reltio if the geerl term is give s reltio of its precedig d succeedig terms i the sequece together with some iitil coditio. Let t > d let { } t > Also t t+ t + t t e defied y t t t + t > t t t > t + This implies tht t is mootoiclly decresig. Sice t > t is ouded elow t is coverget. Let us suppose limt t The lim t+ lim t ( ) lim limt t t t t t t ( t ) t Let { } > + t ; t t t t+ S e defied y S S + + ; d S >. It is cler tht S > d S > S d S+ S ( S + ) ( S + ) S S ( S+ + S)( S+ S) S S S S S+ S S+ + S Sice S + S > + Therefore S + S d S S hve the sme sig. i.e. S > + S if d oly if S > S d S < + S if d oly if S < S.

10 Sequeces d Series - - But we ow tht S > S therefore S3 > S, S4 > S3, d so o. This implies the sequece is icresig sequece. S S S + + S S + S Also ( ) ( S S ) Sice S >, therefore S is the root (+ive) of the S S Te this vlue of S s where The lim( S+ ) lim( S + ) s s+ s s For equtio x + x+ c The product of roots is β c the other root of equtio is therefore Sice S > + S c i.e. the other root β Also ( S ) S + S+ S > S + > or ( S ) S < which shows tht S is ouded d hece it is coverget. Suppose lims s Which shows tht is the limit of the sequece. Theorem Every Cuchy sequece of rel umers hs coverget susequece. S is Cuchy sequece. Let ε > the positive iteger such tht ε S S <,,,,3,... Suppose { } Put ( S S ) ( )... ( ) + S S + + S S ( S S ) ( )... ( ) + S S + + S S ( S S ) + ( S S ) ( S S ) ε ε ε < ε < ε { } is coverget S S S + S Where coverget. ( ) S is certi fix umer therefore { } ε ε S which is susequece of { S } is

11 Sequeces d Series - - Theorem (Cuchy s Geerl Priciple for Covergece) A sequece of rel umer is coverget if d oly if it is Cuchy sequece. Necessry Coditio S e coverget sequece, which coverges to s. The for give ε > positive iteger, such tht ε S s < > Now for > m> S Sm S s+ Sm s S s + S s Let { } m ε ε < + ε Which shows tht { S } is Cuchy sequece. Sufficiet Coditio S is Cuchy sequece the for ε >, positive Let us suppose tht { } iteger m such tht Sice { } ε S Sm < m, > m.. (i) S is Cuchy sequece therefore it hs susequece { S } covergig to s (sy). positive iteger m such tht ε S s < > m.. (ii) Now S s S S + S s S S + S s which shows tht { } Let { } ε ε < + ε > mx( m, m) S is coverget sequece. S e defie y < < S < S < d lso S+ S S, >. (i) Here S >, d < S < Let for some > < S < the < S < S S ( S+ ) < S+ SS i.e. < S+ < < S+ < < S < N S > S+ S + > + S +

12 Ad S + S+ + > S+ S + S+ + > S + is replce y S S+ S S S S S+ SS S S S ( ) S S S S S + S + S + < S S + S S < S S + + < S S + 3 < S S 3 + Sequeces d Series - - S S < S + S S S S < ( ) + Te r + < The for > m we hve S S S S + S S S S m m+ m This implies tht { } t e defied y Let { } S S + S S S S m+ m 3 m ( ) < r + r r ( ) ε S is Cuchy sequece, therefore it is coverget. t For m, N, > m we hve t tm m+ m+ > ( m) m I prticulr if m the t tm > t is ot Cuchy sequece therefore it is diverget. This implies tht { }

13 Theorem (ested itervls) I Sequeces d Series Suppose tht { I } is sequece of the closed itervl such tht I [, ] I, d ( ) I + poit. s the, cotis oe d oly oe Sice I + I < < 3 <... < < < < <... < 3 < < { } is icresig sequece, ouded ove y d ouded elow y. is decresig sequece ouded elow y d ouded ove y. oth re coverget. Ad { } { } Suppose { } d { } coverges to d { } + + But coverges to. + + s. d < <. Theorem (Bolzo-Weierstrss theorem) Every ouded sequece hs coverget susequece. Let { } S e ouded sequece. Te if S d sup S The < S <. Now isect itervl [, ] such tht t lest oe of the two su-itervls cotis ifiite umers of terms of the sequece. Deote this su-itervl y [, ]. If oth the su-itervls coti ifiite umer of terms of the sequece the choose the oe o the right hd. The clerly <. Suppose there exist suitervl [, ] such tht... <... ( ) ( ) Bisect the itervl [, ] i the sme mer d choose [ +, + ] to hve... + < +... d + + ( + ), such tht This implies tht we oti sequece of itervl [ ] ( ) s. we hve uique poit s such tht s [, ] there re ifiitely my terms of the sequece whose legth is ε > tht coti s. For ε there re ifiitely my vlues of such tht S s < Let e oe of such vlue the S s <

14 Agi choose > such tht S s < Cotiuig i this mer we fid sequece { } tht < + d S Sequeces d Series S for ech positive iteger such s <,,3,... Hece there is susequece { S } which coverges to s. Limit Iferior of the sequece Suppose { S } is ouded the we defie limit iferior of { } lim( if S ) limu where U if { S : } ( ) If S is ouded elow the lim if S Limit Superior of the sequece S s follow Suppose { S } is ouded ove the we defie limit superior of { } lim( sups ) limv where V if { S : } If S is ot ouded ove the we hve lim( sup S ) + S s follow Note: (i) A ouded sequece hs uique limit iferior d superior (ii) Let { S } cotis ll the rtiol umers, the every rel umer is susequecil limit the limit superior of S is + d limit iferior of S is (iii) Let { S } ( ) + the limit superior of S is d limit iferior of S is. U if S : (iv) Let { } if + cos π, + cos( + ) π, + cos( + ) π, cos π if is odd + cos( + ) π if is eve + lim if S limu ( ) Also V sup { S : } + cos( + ) π + + cos π lim if S limv ( ) if is odd if is eve

15 Theorem S is coverget sequece the If { } Sequeces d Series ( ) ( ) lims lim if S lim sups Let lims s the for rel umer ε >, positive iteger such tht S s < ε.. (i) i.e. s ε < S < s+ ε V sup S : If { } The ε < < + ε s V s s ε < limv < s+ ε. (ii) from (i) d (ii) we hve s lim sup { S } We c hve the sme result for limit iferior of { } U if { S : } S y tig

16 Ifiite Series Give sequece { }, we use the ottio sum 3... Sequeces d Series or simply i d clled ifiite series or just series. The umers S If the sequece { } re clled the prtil sum of the series. to deotes the S coverges to s, we sy tht the series coverges d write s, the umer s is clled the sum of the series ut it should e clerly uderstood tht the s is the limit of the sequece of sums d is ot otied simply y dditio. S diverges the the series is sid to e diverge. If the sequece { } Note: The ehviors of the series remi uchged y dditio or deletio of the certi terms Theorem If coverges the lim Let S Te lims s Sice S S lim lim S S Therefore ( ) lims s s lims. Note: The coverse of the ove theorem is flse Cosider the series. We ow tht the sequece { S } where S is diverget therefore is diverget series, lthough lim. This implies tht if lim, the is diverget. It is ow s sic diverget test. Theorem (Geerl Priciple of Covergece) A series is coverget if d oly if for y rel umer ε >, there exists positive iteger such tht i < ε > m > i m+ Let S the { S } is coverget if d oly if for ε > positive iteger such tht

17 Sequeces d Series If x < the Ad if x the Theorem Let S S ε m < > m> S S < ε x i m+ x i m x is diverget. e ifiite series of o-egtive terms d let { } its prtil sums the uouded. Sice is coverget if { } S e sequece of S is ouded d it diverges if { } S S + > S S is mootoic icresig d hece it is coverges if therefore the sequece { } { } S is ouded d it will diverge if it is uouded. Hece we coclude tht { S } is uouded. is coverget if { } S is S is ouded d it diverget if Theorem (Compriso Test) Suppose d re ifiite series such tht >, >. Also suppose tht for fixed positive umer λ d positive iteger, < λ The coverges if is coverges d is diverges if is diverges. Suppose is coverget d < λ. (i) the for y positive umer ε > there exists such tht ε i < m i m+ λ > > from (i) < λ < ε Now suppose, > m > i i i m+ i m+ is coverget. is diverget the { } rel umer β > such tht > λβ, > m i i m+ from (i) > > β i i, > m i m+ λ i m+ is coverget. S is uouded.

18 We ow tht is diverget d is diverget s is diverget. The series Sequeces d Series is coverget if > d diverges if. Let S If > the S < S d < ( ) Now S ( ) ( ) 4 6 ( ) ( ) 3 ( ) < S 4 ( ) replcig 3 y, 5 y 4 d so o S ( ) + S S + + S + S S < S < S + S S < + S S < i.e. S < S < S < S < { S } is ouded d lso mootoic. Hece we coclude tht coverget whe >. If the is diverget therefore is diverget whe. is

19 Sequeces d Series Theorem Let >, > d lim λ the the series d ehve lie. Sice lim λ λ < ε. λ Use ε λ λ <. λ λ λ < < λ + λ 3λ < < the we got 3λ < d < λ Hece y compriso test we coclude tht together. x To chec si si x d te 3 x the si d diverges or coverges cosider x si si x x x Applyig limit s coverge or diverge x x si si lim lim x x lim x () x x x d hve the similr ehvior fiite vlues of x except x. Sice is coverget series therefore the give series is lso coverget for 3 fiite vlues of x except x.

20 Sequeces d Series - - Theorem ( Cuchy Codestio Test ) Let, > +, the the series d coverges or diverges together. Let us suppose S d... t d < < S < S < S for > the S ( ) ( )... ( ) ( ) ( )... (... ) < < t S < t S < t < S. (i) Now cosider S ( ) ( )... (... ) > + + ( 4 + 4) + ( ) ( ) ( ) S > t (ii) S > t From (i) d (ii) we see tht the sequece S d t re either oth ouded or oth uouded, implies tht d coverges or diverges together. Cosider the series p If p the lim p therefore the series diverges whe p. If p > the the codestio test is pplicle d we re led to the series Now () p p p ( p ) ( p ) ( ) p < iff p < i.e. whe p >

21 Sequeces d Series - - Ad the result follows y comprig this series with the geometric series hvig commo rtio less th oe. p The series diverges whe ( i.e. whe p ) The series is lso diverget if < p <. If p >, coverges d (l ) p If p the series is diverget. { l } is icresig l decreses d we c use the codestio test to the ove series. We hve p l ( ) l ( ) p we hve the series p p p ( l) l ( ) ( l) which coverges whe p > d diverges whe p. Cosider l Sice { l } is icresig there l decreses. Ad we c pply the codestio test to chec the ehvior of the series l l so l l sice > d is diverges therefore the give series is lso diverges. p

22 Sequeces d Series - - Altertig Series A series i which successive terms hve opposite sigs is clled ltertig series. + ( ) e.g is ltertig series. 3 4 Theorem (Altertig Series Test or Leiiz Test) e decresig sequece of positive umers such tht lim the the Let { } ltertig series + ( ) coverges. Looig t the odd umered prtil sums of this series we fid tht S+ ( ) + ( 3 4) + ( 5 6) ( ) + + Sice { } is decresig therefore ll the terms i the prethesis re o-egtive S + > Moreover S+ 3 S S+ ( + + 3) Sice therefore S+ 3 S+ Hece the sequece of odd umered prtil sum is decresig d is ouded elow y zero. (s it hs +ive terms) It is therefore coverget. Thus S + coverges to some limit l (sy). Now cosider the eve umered prtil sum. We fid tht S+ S+ + d lims lim S ( ) lims lim + + l l lim so tht the eve prtil sum is lso coverget to l. oth sequeces of odd d eve prtil sums coverge to the sme limit. Hece we coclude tht the correspodig series is coverget. Asolute Covergece is sid to coverge solutely if coverges. Theorem A solutely coverget series is coverget. : If is coverget the for rel umer the series i < i < ε m, > i m+ i m+ ε >, positive iteger such tht is coverget. (Cuchy Criterio hs ee used) Note The coverse of the ove theorem does ot hold. ( ) + e.g. is coverget ut is diverget.

23 Theorem (The Root Test) The limsup p coverges solutely if Let Sequeces d Series p < d it diverges if p >. Let p < the we c fid the positive umer ε > such tht p + ε < < p+ ε < > ( ) < p+ ε < ( p + ε ) is coverget ecuse it is geometric series with r <. is coverget coverges solutely. Now let p > the we c fid umer ε > such tht p ε >. > p+ ε > > for ifiitely my vlues of. lim is diverget. Note: The ove test give o iformtio whe p. e.g. Cosider the series d. For ech of these series p, ut is diverget d is coverget. Theorem (Rtio Test) The series + (i) Coverges if limsup < + (ii) Diverges if > for, where is some fixed iteger. If (i) holds we c fid β < d iteger N such tht I prticulr + < β for N N+ < N β < β N+ N N+ β N+ β N < < < β N < β N p N+ p N

24 Sequeces d Series N < β N we put N p N < N β β for N. β +. i.e. is coverget ecuse it is geometric series with commo rtio <. Therefore is coverget (y compriso test) Now if + for the lim is diverget. Note The owledge + implies othig out the coverget or diverget of series. Cosider the series Also ( ) < + with > lim lim lim + lim + e e e e e e the series is diverget. Theorem (Dirichlet) Suppose tht { S }, S is ouded. Let { } e positive term decresig sequece such tht lim, the is coverget. { S } is ouded positive umer λ such tht < λ. S The ( ) i i Si Si i for i S i i Si i S S + S S i i i i i i+ i i+ >

25 ( ) S S + S i i i+ i i i i+ Sequeces d Series ( ) ( ) S S S { } i i i i i+ m m+ + i m+ i m+ is decresig ( ) S S + S i i i i i+ m m+ + i m+ i m+ { ( )} < S + S + S i i i+ m m+ + i m+ { λ( i i+ ) } λm+ λ+ Si i m+ < λ ( i i+ ) m+ + i m+ λ (( m+ + ) + m+ + + ) λ ( m + ) i i ε where ε λ ( m+ ) i m+ < The Theorem Suppose tht certi umer < λ is coverget. ( We hve use Cuchy Criterio here. ) is coverget d tht { } the is lso coverget. Suppose { } is decresig d it coverges to. Put c c d limc is mootoic coverget sequece is coverget { S }, S is coverget It is ouded c is ouded. c + d c d re coverget. is coverget. Now if { } is icresig d coverges to the we shll put c. ( l ) is coverget if > d diverget if. To see this we proceed s follows ( l ) Te ( l ) ( ) l ( l) ( ) l

26 ( l) Sequeces d Series ( ) Sice is coverget whe > d coverges to. Therefore is coverget is lso coverget. Now is diverget for therefore We hve To chec ( ) is coverget or diverget. l l Te ( ) (l) l is diverget lthough therefore is diverget. is diverget. ( ) The series lso diverget if. i.e. it is lwys diverget. is decresig for > d it diverges for. ( l) l ( ) ( ) is decresig, tedig to zero for > Refereces: () Lectures (3-4) Prof. Syyed Gull Shh Chirm, Deprtmet of Mthemtics. Uiversity of Srgodh, Srgodh. () Boo Priciples of Mthemticl Alysis Wlter Rudi (McGrw-Hill, Ic.) Mde y: Atiq ur Rehm (mthcity@gmil.com) Aville olie t i PDF Formt. Pge Setup: Legl ( 8 4 ) Prited: Octoer, 4. Updted: Octoer, 5