i If s n is an unbounded sequence then either s n or sn j If s n is a bounded

I if s n is an unbounded sequence then either s n or

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(i) If (sn) is an unbounded sequence, then eithersn+orsn→ -∞.(j) If(sn)is a bounded sequence andα= sup{sn},then(sn)has asubsequence which converges toα.(k) For any given sequence(sn),(sn)has a convergent subsequence(snk). 4
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(l) For any bounded sequence(sn),lim supsn= sup{sn}.(m) Itαis a subsequential limit of a bounded sequence(sn),thenαlim supsn(n) If every subsequence of a sequence(sn)is convergent, then(sn)itselfmust be convergent. 5
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(o) If(sn)is a divergent sequence, then some subsequence of(sn)mustdiverge.(p) If(sn)is unbounded above, then(sn)has an increasing subsequence(snk)which diverges to +. 6
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2. Let(sn)be a positive sequence such thatlimn→∞sn+1sn=L >1.Prove thatsn+.Hint:limn→∞xn= +for any numberxsuch thatx >1. 7
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3.Limits:Determine the convergence or divergence of each of the followingsequences. If the limit exists, state what it is. (Hint: Use the previous problem,and Theorem 3 from Section 17.)(a)sn=n33n(b)sn=n!n10(c)sn=nnn! 8
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4. For each of the following sequences, determine: (1) Whether the sequence con-verges; if it does, give the limit. (2) If the sequence does not converge, find theset of subsequential limits (if any) and givelim supandlim inf.(a)sn=1n+ 1,nodd32n-1,neven(b)sn=1n,n= 3,6,9,· · ·nn+ 1,n= 1,4,7,10,· · ·n2,n= 2,5,8,11,· · ·(c)sk=ksin24(d)sn=12n+ cos6 9
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