5. Mark True or False. Justify each answer. (you may use the limit laws if neede

Question

5. Mark True or False. Justify each answer. (you may use the limit laws if needed) (a) If {an} converges to a and an > 0 for all n N then a > 0. (b) If {an} and {bn} are both divergent sequences, then {an + bn} diverges. (c) If {an} and {bn} are both divergent sequences, then {anbn} diverges. (d) If {an} and {an + bn} are both convergent sequences, then {bn} converges.

6. Find the following limits (a) limn 3n 2 + 4n 7n2 5n (b) limn sin n 2n + 1 (c) limn ( n2 + 1 n).

8. Determine whether each series converges or diverges. Justify your answer. (a) X n=1 n 5 2 n (b) X n=1 2 n n! (c) X n=1 1 (3n 2)(3n + 1) (d) X n=1 ( n + 1 n)

9. Mark each statement True or False. Justify your answer. (a) P n=1 an converges iff limn an = 0. (b) The geometric series P n=0 r n converges iff r < 1. (c) P n=1 an converges if the sequence {an} is bounded. (d) P n=1 an converges if the sequence {sn} where sn = a1 + ... + an is bounded.

10. For each subset of R, find its maximum and supremum, minimum and infimum, if they esist. Otherwise write ”none”. (a) [0, 2) (b) {r Q : 3 < r2 4} (c) n=1(1 1 n , 1 + 1 n )

11. Let S be a nonempty subset of R and let m = sup S. Prove that m S iff m = max S. 12. Let S be a nonempty bounded subset of R. Show inf S sup S. What can you conclude if inf S = sup S?

Explanation / Answer

PLEASE POST AS SEPARATE QUESTIONS:

5) a) TRUE. Since an>0 for all n thus the convergence must be >0.

b) False: an = n, bn = n, an + bn = 0.

c) False: an = bn = (1)n, anbn = 1

d) True

DO THUMBS UP ^_^

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