which contradicts the assumption s n t n for all n Corollary Suppose t n t Ift

Which contradicts the assumption s n t n for all n

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which contradicts the assumptionsntnfor alln.CorollarySupposetnt. Iftn0for alln,thent0.Infinite LimitsDefinition 3.A sequence{sn}diverges to+(sn+)if to each real numberMthereis a positive integerNsuch thatsn> Mfor alln > N.{sn}diverges to-∞(sn→ -∞)if to each real numberMthere is a positive integerNsuch thatsn< Mfor alln > N.THEOREM 6.Suppose that{sn}and{tn}are sequences such thatsntnfor alln.1. Ifsn+,thentn+.2. Iftn→ -∞,thensn→ -∞.THEOREM 7.Let{sn}be a sequence of positive numbers.Thensn+if and only if1/sn0.Proof:Supposesn→ ∞. Let>0and setM= 1/. Then there exists a positive integerNsuch thatsn> Mfor alln > N. Sincesn>0,1/sn<1/M=for alln > Nwhich implies1/sn0.Now suppose that1/sn0. Choose any positive numberMand let= 1/M. Then thereexists a positive integerNsuch that0<1sn<=1Mfor alln > N.that is,1sn<1M.Sincesn>0for alln,1/sn<1/Mfor alln > Nimpliessn> Mfor alln > N. Therefore,sn→ ∞. Exercises 2.2 1. Prove or give a counterexample.(a) Ifsnsandsn>0for alln,thens >0.(b) If{sn}and{tn}are divergent sequences, then{sn+tn}is divergent.(c) If{sn}and{tn}are divergent sequences, then{sntn}is divergent.(d) If{sn}and{sn+tn}are convergent sequences, then{tn}is convergent.(e) If{sn}and{sntn}are convergent sequences, then{tn}is convergent.(f) If{sn}is not bounded above, then{sn}diverges to +.2. Determine the convergence or divergence of{sn}. Find any limits that exist. 17
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(a)sn=3-2n1 +n(b)sn=(-1)nn+ 2(c)sn=(-1)nn2n-1(d)sn=23n32n(e)sn=n2-2n+ 1(f)sn=1 +n+n21 + 3n3. Prove the following:(a)limn→∞n2+ 1-n= 0.(b)limn→∞n2+n-n=12.4. Prove Theorem 4.5. Prove Theorem 6.6. Let{sn},{tn},and{un}be sequnces such thatsntnunfor alln. Prove that ifsnLandunL,thentnL.II.3. MONOTONE SEQUENCES AND CAUCHY SEQUENCES Monotone Sequences Definition 4.A sequence{sn}isincreasingifsnsn+1for alln;{sn}isdecreasingifsnsn+1for alln. A sequence ismonotoneif it is increasing or if it is decreasing. Examples (a) 1 , 1 2 , 1 3 , 1 4 , . . ., 1 n , . . . is a decreasing sequence. (b) 2 , 4 , 8 , 16 , . . ., 2 n , . . . is an increasing sequence. (c) 1 , 1 , 3 , 3 , 5 , 5 , . . ., 2 n - 1 , 2 n - 1 , . . . is an increasing sequence. (d) 1 , 1 2 , 3 , 1 4 , 5 , . . . is not monotonic. Some methods for showing monotonicity: (a) To show that a sequence is increasing, show that s n +1 s n 1 for all n . For decreasing, show s n +1 s n 1 for all n . The sequence s n = n n + 1 is increasing: Since s n +1 s n = ( n + 1) / ( n + 2) n/ ( n + 1) = n + 1 n + 2 · n + 1 n = n 2 + 2 n + 1 n 2 + 2 n > 1 18
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