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Unformatted text preview: a bounded subsequence. 4 3. Give the deﬁnition of a Cauchy sequence and use the deﬁnition to show that ( π n ) is not a Cauchy sequence. Deﬁnition: Proof that ( π n ) is not a Cauchy sequence: 5 4. Complete the following sentences by inserting the words convergent or divergent (do not explain why): 4a. ∑ 1 / √ n is 4b. ∑ 1 /n 2 is 4c. ∑ (1) n (1 / √ n ) is 4d. ∑ (1) n +1 (1 /n p ) with p > 1 is 6 5. Compute the two limits: ( 5a ) lim n →∞ (2 + 1 n ) n 2 n +2 (You may use without proof that the Euler number e = lim(1+1 /n ) n . ) ( 5b ) lim x → √ 1 + x√ 1x xx 2 (You may use without proof that, if lim f = c ≥ , then lim √ f = √ lim f ) . 7 6. Show, using ( ±, δ ) , that for every c ∈ R lim x → c x 3 = c 3 . 8 9 10...
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 Fall '10
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 Topology, Mathematical analysis, Cauchy sequence, subsequence, 40pts

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