CSIR NET - MATHEMATICAL SCIENCE

1 csir net , GATE, IIT-JAM, UGC NET , TIFR, IISc, JEST, JNU, BHU, ISM, IBPS, CSAT, SLET, NIMCET, CTET Phone: 0744-2429714 Mobile: 9001297111, 9829567114, 9001297243 Website: E-Mail: Address: 1-C-8, Sheela Chowdhary Road, SFS, TALWANDI, KOTA, RAJASTHAN, 324005 Page 1 For IIT-JAM, JNU, GATE, NET, NIMCET and Other Entrance Exams1-C-8, Sheela Chowdhary Road, Talwandi, Kota (Raj.) Tel No. 0744-2429714 Web Site csir net - MATHEMATICAL SCIENCESAMPLE THEORY SEQUENCES , SERIES AND LIMIT POINTS OF SEQUENCESSEQUENCESLIMITS : INFERIO R & SUPERIOR SEQUENCE TESTS FOURIER SERIESSOME PROBLEMSALGEBRA OF SEQUENCES csir net , GATE, IIT-JAM, UGC NET , TIFR, IISc, JEST, JNU, BHU, ISM, IBPS, CSAT, SLET, NIMCET, CTET Phone: 0744-2429714 Mobile: 9001297111, 9829567114, 9001297243 Website: E-Mail: Address.

2 1-C-8, Sheela Chowdhary Road, SFS, TALWANDI, KOTA, RAJASTHAN, 324005 Page 2 SEQUENC E A sequence in a set S i s a function whose domain is the set N of natural numbers and whose range i s a subset of S. A sequence whose range i s a subset of R i s call ed a real sequence. Sn = u1 + u2 + u3 + .. + un S1 = u1 S2 = u1 + u2 S3 = u1 + u2 + u3 Sn = u1 + u2 + u3 + .. + un seri es Sequence Bounded Sequence: A sequence i s said to be bounded if and onl y i f i ts range i s bounded. Thus a sequence Sn i s bounded if there exi sts nkSK,n N []nSk,K T he l. u. b (Suprem um ) and the .b (infim um) of the range of a bounded sequence may be referred as i ts and l.

3 Respecti vel y. Limits inferior and Superior: From the defi ni tion of li mi t i n Secti on , i t follows that the limi ti ng behavi or of any sequence {an } of real num bers, depends onl y on sets of the form {an : n m }, i .e., {am, am + 1, am + 2, .. }. In thi s regard we make the foll owi ng defi ni ti on. Definition: Let {an } be a sequence of real numbers (not necessari l y bounded). We defi ne nlim i nf an = nsup i nf {an, an + 1, an + 2,.. } csir net , GATE, IIT-JAM, UGC NET , TIFR, IISc, JEST, JNU, BHU, ISM, IBPS, CSAT, SLET, NIMCET, CTET Phone: 0744-2429714 Mobile: 9001297111, 9829567114, 9001297243 Website: E-Mail: Address: 1-C-8, Sheela Chowdhary Road, SFS, TALWANDI, KOTA, RAJASTHAN, 324005 Page 3 And nlim sup an = ninf sup {an, an + 1, an + 2.}

4 } As the l imi t inferi or and limi t superi or respectivel y of the sequence {an }. We shal l denote li mit i nferior and li mi t superior of {an } by nlim an and nl im an or si m pl y by lim an and l im an respecti vel y. We shall use the foll owi ng notations for the sequence {an }, for each n N nA = inf {an, an + 1, an + 2, .. }, And nA = sup {an, an + 1, an + 2, .. }. T herefore, we have l im an = nnsup A And l im an = ni nf An Now {an + 1 , an + 2, ..} {an, an + 1, an + 2, ..}, Therefore by taki ng i nfim um and suprem um respecti vel y, it follows that n 1nAA+ And n 1nAA+ T hi s i s true for each n N. T he above i nequal ities show that the associated sequences {nA } and {nA } m onotoni call y i ncrease and decrease respecti vel y with n.

5 Remark: It should be noted that both lim its i nferi or and superior exi st uniquel y (fi ni te or i nfinite) for all real sequences. Theorem: If {an } i s any sequence, then l im ( an ) = lim , and l im ( an ) = lim an. Let bn = an, n N then we have nB = inf {bn, bn + 1, ..} = sup {an, an + 1, ..} = nA And so lim ( an ) lim = bn = sup()12B ,B ,.. csir net , GATE, IIT-JAM, UGC NET , TIFR, IISc, JEST, JNU, BHU, ISM, IBPS, CSAT, SLET, NIMCET, CTET Phone: 0744-2429714 Mobile: 9001297111, 9829567114, 9001297243 Website: E-Mail: Address: 1-C-8, Sheela Chowdhary Road, SFS, TALWANDI, KOTA, RAJASTHAN, 324005 Page 4 = sup{}12A , A.

6 = i nf {}12A ,A ,.. = i nf nA = lim an. Al so, l im an = l im ( (an )) = l im ( an ). Theorem: If {an } i s any sequence, then l im an = i f and onl y i f {an } i s not bounded below, And l im an = + if and onl y i f {an } i s not bounded above. Let nA = inf {an, an + 1, ..}, And nA = sup {an, an + 1, ..}, n N By defini ti on we have l im an = sup {}12A , A ,.. = nA = , n N i nf {an, an + 1, .. } = , n N {an } is not bounded bel ow: T he proof for li mi t superior i s si mil ar. Corollary: If {an } i s any sequence, then (i ) < li m an + i ff {an } i s bounded below.

7 And (i i ) li man < + i ff {an } is bounded above. For bounded sequences, we have the fol lowing useful cri teria for lim its i nferior and superi or respecti vel y. Limit points of a sequence: csir net , GATE, IIT-JAM, UGC NET , TIFR, IISc, JEST, JNU, BHU, ISM, IBPS, CSAT, SLET, NIMCET, CTET Phone: 0744-2429714 Mobile: 9001297111, 9829567114, 9001297243 Website: E-Mail: Address: 1-C-8, Sheela Chowdhary Road, SFS, TALWANDI, KOTA, RAJASTHAN, 324005 Page 5 A number is sai d to be a l imi t point of a sequence Sn i f gi ven any nbd of , Sn belongs to the sam e for an infi ni te num ber of values of n.

8 Now {Sn+ 1 Sn+2, Sn+ 3, ..} {Sn, Sn+ 1, Sn+ 2, ..}, therefore by taking i nfimum and supremum respecti vel y, i f fol lows that n 1nAA+ and n 1nAA+ for each n N Remark: Both li mi ts inferi or and superior exist uni quel y (fi ni te or infi ni te) for all real sequence. Theorem: If {Sn} i s any sequence, then i nf nnnSli m SSup S If {Sn} i s any sequence, then {}nnl im Slim S = And {}nnl imSlim S = Some Important Properties of Algebra of sequences 1. If {an} i s a bounded sequence such that an > 0 for all n N, then (i ) nnn11lim,i f lim a0ali ma => (i i ) nnn11lim,if lim a0alim a => 5. If {an} and {bn} are bounded sequence, nna0, b0 > for all n N, then (i ) nnnnnalim alim,if lim b0blim b > (i i ) nnnnnalim alim,if l im b0blim b > SOME IMPORTANT SEQUENCE TESTS 1.

9 Cauchy s root test Let un be +ve term seri es and { }nunnlim u = csir net , GATE, IIT-JAM, UGC NET , TIFR, IISc, JEST, JNU, BHU, ISM, IBPS, CSAT, SLET, NIMCET, CTET Phone: 0744-2429714 Mobile: 9001297111, 9829567114, 9001297243 Website: E-Mail: Address: 1-C-8, Sheela Chowdhary Road, SFS, TALWANDI, KOTA, RAJASTHAN, 324005 Page 6 T hen the series i s (i ) Cgt i f < 1 (i i ) Dgt i f > 1 (i ii ) No fi rm deci si on is possi ble i f = 1 2. Raabe s test Let un be a +ve term seri es and nn 1ul imn1u+ = then the seri es i s (i ) Cgt i f > 1 (i i ) Dgt i f < 1 (i ii ) No fi rm deci si on is possi ble i f = 1 3.

10 Logarithmic Test : If un i s +ve term s seri es such that nnn 1ulim nlogu + = T hen the series (i ) cgt i f > 1 (i i ) dgt i f < 1 4. Absolute convergent A seri es un is sai d to be absolutel y cgt i f the posi ti ve term seri es |un| formed by the m odul e of the term s of the series i s convergent. 5. Conditional convergent A seri es i s sai d to be condi ti onall y convergent i f i t i s convergent wi thout being absolutel y convergent. Theorem: Every absolute convergent series i s convergent. csir net , GATE, IIT-JAM, UGC NET , TIFR, IISc, JEST, JNU, BHU, ISM, IBPS, CSAT, SLET, NIMCET, CTET Phone: 0744-2429714 Mobile: 9001297111, 9829567114, 9001297243 Website: E-Mail: Address: 1-C-8, Sheela Chowdhary Road, SFS, TALWANDI, KOTA, RAJASTHAN, 324005 Page 7 Note.