CSIR NET - MATHEMATICAL SCIENCE

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1 CSIR NET - MATHEMATICAL SCIENCE SAMPLE THEORY SEQUENCES, SERIES AND LIMIT POINTS OF SEQUENCES SEQUENCES LIMITS : INFERIOR & SUPERIOR ALGEBRA OF SEQUENCES SEQUENCE TESTS FOURIER SERIES SOME PROBLEMS For IIT-JAM, JNU, GATE, NET, NIMCET ad Other Etrace Exams -C-8, Sheea Chowdhary Road, Tawadi, Kota (Raj.) Te No Web Site E-mai-vmcasses@yahoo.com Phoe: Mobie: 997, , Website: E-Mai: vmcasses@yahoo.com /ifo@vmcasses.com Address: -C-8, Sheea Chowdhary Road, SFS, TALWANDI, KOTA, RAJASTHAN, 345 Page

2 SEQUENCE A seqece i a set S is a fctio whose domai is the set N of atra mbers ad whose rage is a sbset of S. A seqece whose rage is a sbset of R is caed a rea seqece. S S S + S S series Seqece Boded Seqece: A seqece is said to be boded if ad oy if its rage is boded. Ths a seqece S is boded if there exists k S K, N S [ k,k] The.. b (Sremm) ad the g..b (ifimm) of the rage of a boded seqece may be referred as its g..b ad..b resectivey. Limits iferior ad Serior: From the defiitio of imit i Sectio.4, it foows that the imitig behavior of ay seqece {a } of rea mbers, deeds oy o sets of the form {a : m}, i.e., {a m, a m +, a m +,... }. I this regard we make the foowig defiitio. Defiitio: Let {a } be a seqece of rea mbers (ot ecessariy boded). We defie im if a s if {a, a +, a +,... } Phoe: Mobie: 997, , Website: E-Mai: vmcasses@yahoo.com /ifo@vmcasses.com Address: -C-8, Sheea Chowdhary Road, SFS, TALWANDI, KOTA, RAJASTHAN, 345 Page

3 Ad im s a if s {a, a +, a +,... } As the imit iferior ad imit serior resectivey of the seqece {a }. We sha deote imit iferior ad imit serior of {a } by im a resectivey. im a ad im We sha se the foowig otatios for the seqece {a }, for each N Ad Therefore, we have A if {a, a +, a +,... }, A s {a, a +, a +,... }. im a s A a or simy by im a ad Ad im a if A Now {a +, a +,....} {a, a +, a +,....}, Therefore by takig ifimm ad sremm resectivey, it foows that A This is tre for each N. + A Ad A + A T he above ieqaities show that the associated seqeces { A } ad { A } mootoicay icrease ad decrease resectivey with. Remark: It shod be oted that both imits iferior ad serior exist iqey (fiite or ifiite) for a rea seqeces. Theorem: If {a } is ay seqece, the im ( a ) im, ad im ( a ) im a. Let b a, N the we have Ad so B if {b, b +,....} s {a, a +,....} A im ( a ) im b s( B,B,... ) Phoe: Mobie: 997, , Website: E-Mai: vmcasses@yahoo.com /ifo@vmcasses.com Address: -C-8, Sheea Chowdhary Road, SFS, TALWANDI, KOTA, RAJASTHAN, 345 Page 3

4 s{ A, A,... } if { A,A,... } if A im a. Aso, im a im ( (a )) im ( a ). Theorem: If {a } is ay seqece, the im a if ad oy if {a } is ot boded beow, Ad Let im a + if ad oy if {a } is ot boded above. A if {a, a +,....}, Ad By defiitio we have A s {a, a +,....}, N im a s { A, A,... } A, N if {a, a +,... }, N {a } is ot boded beow: The roof for imit serior is simiar. Coroary: If {a } is ay seqece, the ad (i) < im a + iff {a } is boded beow. (ii) im a < + iff {a } is boded above. For boded seqeces, we have the foowig sef criteria for imits iferior ad serior resectivey. Limit oits of a seqece: Phoe: Mobie: 997, , Website: E-Mai: vmcasses@yahoo.com /ifo@vmcasses.com Address: -C-8, Sheea Chowdhary Road, SFS, TALWANDI, KOTA, RAJASTHAN, 345 Page 4

5 A mber ξ is said to be a imit oit of a seqece S if give ay bd of ξ, S beogs to the same for a ifiite mber of vaes of. Now {S + S +, S +3,...} {S, S +, S +,...}, therefore by takig ifimm ad sremm resectivey, if foows that A + A ad A + A for each N Remark: Both imits iferior ad serior exist iqey (fiite or ifiite) for a rea seqece. Theorem: If {S } is ay seqece, the if S im S S S If {S } is ay seqece, the { } im S ims Ad { } im S ims Some Imortat Proerties of Agebra of seqeces. If {a } is a boded seqece sch that a > for a N, the (i) im,if ima > a ima (ii) im,if ima > a im a 5. If {a } ad {b } are boded seqece, a, b > for a N, the a ima (i) im,if im b > b im b a ima (ii) im,if im b > b im b SOME IMPORTANT SEQUENCE TESTS. Cachy s root test Let Σ be +ve term series ad im { } Phoe: Mobie: 997, , Website: E-Mai: vmcasses@yahoo.com /ifo@vmcasses.com Address: -C-8, Sheea Chowdhary Road, SFS, TALWANDI, KOTA, RAJASTHAN, 345 Page 5

6 The the series is (i) Cgt if < (ii) Dgt if > (iii) No firm decisio is ossibe if. Raabe s test Let Σ be a +ve term series ad im + the the series is (i) Cgt if > (ii) Dgt if < (iii) No firm decisio is ossibe if 3. Logarithmic Test: If Σ is +ve terms series sch that im og + The the series (i) cgt if > (ii) dgt if < 4. Absote coverget A series Σ is said to be absotey cgt if the ositive term series Σ formed by the mode of the terms of the series is coverget. 5. Coditioa coverget A series is said to be coditioay coverget if it is coverget withot beig absotey coverget. Theorem: Every absote coverget series is coverget. Phoe: Mobie: 997, , Website: E-Mai: vmcasses@yahoo.com /ifo@vmcasses.com Address: -C-8, Sheea Chowdhary Road, SFS, TALWANDI, KOTA, RAJASTHAN, 345 Page 6

7 Note. (i) If Σ i s cgt withot beig absotey cgt. I.e. if Σ is coditioay cgt the each of the +ve term series Σg() ad Σh() diverges to ifiity which foows from g( ) + h ( ) (ii) It shod be oted that three are o comariso tests for the cgt of coditioay cgt series. Ateratig series A series whose terms are ateratey +ve ad ve is caed a ateratig series. 6. Leibitz s test Let be a seqece sch that N (i) (ii) + (iii) im The ateratig series () () + (3) (4) ( ) + ()... is cgt. 7. Abe s Test If a is a ositive, mootoic decreasig fctio ad if Σ is coverget series, the the series Σ a is aso coverget. Uiform covergece Poit wise Cov ergece of Seqece of Fctios Defiitio: A seqece of fctios {f } defied o [a, b] is said to be oit-wise coverget to a fctio f o [a, b], if to each > to each x [a, b], there exists a ositive iteger m (deedig o ε ad the oit x) sch that f (x) f(x) < ε > m ad x [a,b]. Phoe: Mobie: 997, , Website: E-Mai: vmcasses@yahoo.com /ifo@vmcasses.com Address: -C-8, Sheea Chowdhary Road, SFS, TALWANDI, KOTA, RAJASTHAN, 345 Page 7

8 The fctio f is caed the oit-wise imit of the seqece {f }. We write ( ) ( ) FOURIER SERIES a f(x) + + Where ( < x < π) α a cosx b six π π a ( ) π f x dx π cosx dx a f ( x) Ad b f ( x) π π Ad for ( π < x < π ) π π a ( ) π f x dx a f ( x) π six dx π cosx dx π π six dx Ad b f ( x) π π im f x f x. Where f(x) is a odd fctio; a ad a where f(x) is a eve fctio; b. Forier series i the iterva ( < x < ) is f(x) a πx πx + a cos + b si f x dx Where a ( ) πx f x cos dx a ( ) Ad b ( ) πx f x si dx I the iterva ( < x < ) Phoe: Mobie: 997, , Website: E-Mai: vmcasses@yahoo.com /ifo@vmcasses.com Address: -C-8, Sheea Chowdhary Road, SFS, TALWANDI, KOTA, RAJASTHAN, 345 Page 8

9 + + πx f(x) dx, a f x cos dx a ( ) + πx Ad b f(x)si dx Note: Whe f(x) is a odd fctio, a ad a whe f(x) is a eve fctio, b. Haf-Rage series ( < x < π) A fctio f(x) defied i the iterva < x < π has two distict haf-rage series. (i) The haf-rage cosie series is f(x) a + a cos x π Where a f ( x) dx π ad a ( ) (ii) The haf rage sie series is, f(x) Σb si x Where b ( ) π f x si x dx. π π f x cos x dx Haf-Rage Series ( < x < ) A fctio f (x) defied i the iterva ( < x < ) ad havig two distict haf-rage series. (i) The haf rage cosie series is, f(x) a Where a ( ) πx +Σ a cos f x dx Ad a f ( x) cos πx dx (ii) The haf-rage sie series is, f(x) Σb si πx Phoe: Mobie: 997, , Website: E-Mai: vmcasses@yahoo.com /ifo@vmcasses.com Address: -C-8, Sheea Chowdhary Road, SFS, TALWANDI, KOTA, RAJASTHAN, 345 Page 9

10 Where b ( ) x f x si π dx Comex form of Forier series f(x) + m c e m imx π i mx Where c m f ( x) e + π π π c ( ) π f x dx ad dx C -m +π π π ( ) imx f x e dx. Parseva s Idetity For Forier series, f(x) The Parseva s idetity is a πx πx + a cos + b si, x < < f x dx a b a ( ) ( ) FOURIER INTEGRAL The Forier series of eriodic fctio f (x) o the iterva (, + ) is give by Where a The f (x) a + πx a b cos πx f (x) dx f (t)cos πt dt f (t)si πt dt + b + si πx f(t) dt...() Phoe: Mobie: 997, , Website: E-Mai: vmcasses@yahoo.com /ifo@vmcasses.com Address: -C-8, Sheea Chowdhary Road, SFS, TALWANDI, KOTA, RAJASTHAN, 345 Page

11 f(x) + d f (t )cos(x π This is a form of Forier Itegra. t)dt SOME PROBLEMS. The set of a ositive vaes of a for which the series ta coverges, is a (), 3 (), 3 (3), 3 (4), 3. Match the foowig Series (X) Domai of Covergece (Y) A. x (i) [, ] 3 + x (ii) [ e, + e] B. ( ) + C. ( ) + ( x ) (iii) [, ] D. ( + )! x (iv) [, ] A B C D () (iv) (iii) (ii) (i) () (iv) (iii) (i) (ii) (3) (iii) (iv) (i) (ii) (4) (i) (ii) (iv) (iii) 3. The series is () Coverget, if diverget, if < () Coverget, if > ad diverget, if Phoe: Mobie: 997, , Website: E-Mai: vmcasses@yahoo.com /ifo@vmcasses.com Address: -C-8, Sheea Chowdhary Road, SFS, TALWANDI, KOTA, RAJASTHAN, 345 Page

12 (3) Coverget, if ad diverget, if > (4) Coverget, if < ad diverget, if 4. For the imroer itegra x x α e dx which oe of the foowig is tre () if α <, coverget ad if α, diverget () if α >, Coverget ad if α <, diverget (3) if α >, coverget ad if α <, diverget (4) If α >, diverget ad if α <, coverget 5. Let A R ad Let f f f be fctios o A to R ad Let c be a cster oit of A if L k Lim f k for k,..., The Lim [f(x)] x c c () L () L k k N (3) L (4) x c ANSWER KEY: -. (4),. (), 3. (), 4. (3), 5. (3). (4) Use the foow ig rests: () Let Σa & Σb be two ositive term series (i) If (ii) If a Lt, b a Lt b () The series series beig a fiite o zero costat, the Σa & Σb both coverge or diverge together. & Σβν χονϖεργεσ, the Σa aso coverges. coverges if > & diverges if. We comare the give series with the a a a ta Lt Lt a a ta Phoe: Mobie: 997, , Website: E-Mai: vmcasses@yahoo.com /ifo@vmcasses.com Address: -C-8, Sheea Chowdhary Road, SFS, TALWANDI, KOTA, RAJASTHAN, 345 Page

13 Lt a For this imit to be zero or some other fiite mber 3 i.e. 3 & for the series to be coverget, a > a a > 3 a > 3 a, 3 As. is (D). () (i) x 3 a ;a ( + ) a R im im + a + So the domai of a is [, ] For x the give ower series is Which is coverget. For x the give ower series is Which is coverget, by eibitz s test. As. is (iv) (ii) ( ) + x + a + 3 R im im a + + The iterva of covergece [, ] Phoe: Mobie: 997, , Website: E-Mai: vmcasses@yahoo.com /ifo@vmcasses.com Address: -C-8, Sheea Chowdhary Road, SFS, TALWANDI, KOTA, RAJASTHAN, 345 Page 3

14 for x, the series becomes +... Which is coverget by Leibitz s test 3 5 For x the series becomes Which is agai coverget Hece the exact iterva of covergecy is [, ]. As. is (iii) a (iii) R im im a + Sice the give ower series is abot the oit x the iterva of covergece is + < x < + < x < ( ) + for x +, the give series which is coverget by eibitz s test. Hece the exact iterva of covergece is [, ]. As. is (i) (iv) ( + )! x The give ower series is abot the oit x ( + ) ( + ) + a! R im im a! + + im im + e As. is (ii) The iterva of covergece is [ e, + e], 3. () Negectig the first term ad + ( ) ( )( + ) ( ) ( + ) Phoe: Mobie: 997, , Website: E-Mai: vmcasses@yahoo.com /ifo@vmcasses.com Address: -C-8, Sheea Chowdhary Road, SFS, TALWANDI, KOTA, RAJASTHAN, 345 Page 4

15 or, im + + im + Ratio test fais. og + og + + og + og im og im From Logarithmic test. The series is coverget, if >, i.e., > The series is diverget, if <, i.e., < Phoe: Mobie: 997, , Website: E-Mai: vmcasses@yahoo.com /ifo@vmcasses.com Address: -C-8, Sheea Chowdhary Road, SFS, TALWANDI, KOTA, RAJASTHAN, 345 Page 5

16 The test fais, if i.e., Now og or, og or, og + og 3 4 og + 7 og (3) or, 3 og 7 og im Hece by higher ogarithmic test the give series is diverget, if. Hece the give series is coverget whe > ad diverget whe. The correct aswer is (). α x x e dx, Wheα >, the give itegra is a roer itegra ad hece it is coverget. Whe α <, the itegrad becomes ifiite at x. Now µ α x µ +α x im x.x e im x e x x if µ + α,i.e., µ α We the have < µ < whe < α < ad µ > where α <. It foows byµ -test that the itegra is coverget whe < α < ad diverget whe α <. Ad we have roved above that the itegra is coverget whe α >. Coseqety the give itegra is coverget if α > ad diverget if α <. Phoe: Mobie: 997, , Website: E-Mai: vmcasses@yahoo.com /ifo@vmcasses.com Address: -C-8, Sheea Chowdhary Road, SFS, TALWANDI, KOTA, RAJASTHAN, 345 Page 6

17 5. (3) if L k im fk x c the it foows from a by kow rest which is caed a Idctio armet that L + L + + L im f( + f + + f ), x c ad L L L im(f f f ). I articar, we dedce that if L im x c f ad N, the L im (f(x)). x c Phoe: Mobie: 997, , Website: E-Mai: vmcasses@yahoo.com /ifo@vmcasses.com Address: -C-8, Sheea Chowdhary Road, SFS, TALWANDI, KOTA, RAJASTHAN, 345 Page 7