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Question 3: (b) Let (xn) be a sequence of real numbers such that lim v In = 1. Is it true n->oo or false that in such case the series 00 In diverges? If it is true give a brief n=1 explanation and if it is false give a counterexample. Question 4: Determine whether the following series 80 (-1)nn (b) 3n-1 n=1 converge or diverge. Clearly state any convergence or divergence criteria that you are using. Question 5: Let (xn) and (yn) be sequences in R. We want to show that if [xn < Yn for all n € N and n-1 Yn converges then n-1 In converges as well. For this we suggest the following proof: We have that lim Yn = 0 and that n < yn for all n € N. n->oo Therefore the sequence (|xn]) converges to zero. The latter implies that ,00 |xn converges and thus the series Ln-1 In converges. Determine whether the above proof is correct or wrong. If it is correct, explain the underlined statements. If it is wrong, identify the incorrect claim and explain why it is wrong.

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