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# 319II - a bounded subsequence 4 3 Give the deﬁnition of a...

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MAT 319 SPRING 2008 MIDTERM II !!! WRITE YOUR NAME, STUDENT ID BELOW !!! NAME : ID : THERE ARE SIX (6) PROBLEMS. THEY HAVE THE INDICATED VALUE. SHOW YOUR WORK!!! DO NOT TEAR-OFF ANY PAGE NO CALCULATORS NO PHONES ON YOUR DESK: ONLY test, pen, pencil eraser. 1 40pts 2 40pts 3 40pts 4 40pts 5 50pts 6 40pts Total 250pts 1

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2 !!! WRITE YOUR NAME, STUDENT ID AND LECTURE N. BELOW !!! NAME : ID : LECTURE N. 1. State and prove the Cauchy Convergence Criterion.
3 2. In each case provide an example or state that it is not possible to do so and explain why. 2a Give one example of a sequence that is monotone, bounded and properly divergent. 2b Give one example of a sequence that is positive, bounded with two subsequences converging to 0 and 1 , respectively. 2c Give one example of a sequence that is bounded and admits a divergent subsequence. 2d Give one example of a sequence that is unbounded and it admits

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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-√ 1-x x-x 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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