If s n converges then s n is bounded Proof Suppose s n s There exists a

If s n converges then s n is bounded proof suppose s

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If{sn}converges, then{sn}is bounded. THEOREM 3.Let{sn}and{an}be sequences and suppose that there is a positive numberkand a positive integerNsuch that|sn| ≤k anfor alln > N.Ifan0,thensn0. 14
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Therefore,sn0.CorollaryLet{sn}and{an}be sequences and letsR.Suppose that there is a positivenumberkand a positive integerNsuch that|sn-s| ≤k anfor alln > N.Ifan0,thensns. Exercises 2.1 1. True – False.Justify your answer by citing a theorem, giving a proof, or giving a counter-example.(a) Ifsns,thensn+1s.(b) Ifsnsandtns,then there is a positive integerNsuch thatsn=tnfor alln > N.(c) Every bounded sequence converges(d) If to each>0there is a positive integerNsuch thatn > Nimpliessn<,thensn0.(e) Ifsns,thensis an accumulation point of the setS={s1, s2,· · ·}.2. Prove thatlim3n+ 1n+ 2= 3.3. Prove thatlimsinnn= 0.4. Prove or give a counterexample:(a) If{sn}converges, then{|sn|}converges.(b) If{|sn|}converges, then{sn}converges.5. Give an example of:(a) A convergent sequence of rational numbers having an irrational limit.(b) A convergent sequence of irrational numbers having a rational limit.6. Give the first six terms of the sequence and then give the nthterm(a)s1= 1,sn+1=12(sn+ 1)(b)s1= 1,sn+1=12sn+ 1(c)s1= 1,sn+1= 2sn+ 17. use induction to prove the following assertions:(a) Ifs1= 1andsn+1=n+ 12nsn,thensn=n2n-1.(b) Ifs1= 1andsn+1=sn-1n(n+ 1),thensn=1n. 15
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8. Letrbe a real number,r= 0. Define a sequence{Sn}byS1=1S2=1 +rS3=1 +r+r2...Sn=1 +r+r2+· · ·+rn-1...(a) Supposer= 1. What isSnforSn= 1,2,3, . . .?(b) Supposer= 1. Find a formula forSn.9. Setan=1n(n+ 1), n= 1,2,3, . . .,and form the sequenceS1=a1S2=a1+a2S3=a1+a2+a3...Sn=a1+a2+a3+· · ·+an...Find a formula forSn.II.2. Supposesnsandtnt. Then:1.sn+tns+t.2.sn-tns-t.3.sntnst.Special case:ksnksfor any numberk.4.sn/tns/tprovidedt= 0andtn= 0for alln.THEOREM 5.Supposesnsandtnt. Ifsntnfor alln,thenst. 16
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