 # 1. (a) Let (sn) ∞ n=1 and (tn) ∞ n=1 be non-negativ...

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1. (a) Let (sn) ∞ n=1 and (tn) ∞ n=1 be non-negative bounded sequences. Prove that lim sup(sntn) ≤ (lim sup sn)(lim sup tn) (Hint: Observe that for any fixed N ∈ N, for all n > N we have sn ≤ sup{sk|k > N}. Same thing happens for tn and hence sntn ≤ sup{sk|k > N} sup{tk|k > N}. Use this to prove the result) (b) Give an example of non-negative bounded sequences with lim sup(sntn) < (lim sup sn)(lim sup tn), i.e. the equality does not hold. 2. Determine whether the following series converges or diverges. Justify your claim. (a) X∞ n=1 (−1)n √ n (b) X∞ n=1 2 n √ n! (c) X∞ n=1 n! n4 + 3 (d) X∞ n=1 1 nn (e) X∞ n=1 (−1)n n! 2 n 3. For this question, give an example of or prove that such an example does not exist. Prove all your claims. (a) A convergent series P∞ n=1 an for which P∞ n=1 a 2 n converges. (b) A convergent series P∞ n=1 an for which P∞ n=1 a 2 n diverges. (c) A convergent series P∞ n=1 an with an ≥ 0 for all n ∈ N, for which P∞ n=1 a 2 n diverges. (d) A divergent series P∞ n=1 an for which P∞ n=1 a 2 n converges. (e) A divergent series P∞ n=1 an for which P∞ n=1 a 2 n diverges. 4. Let (an) ∞ n=1 be a sequence such that lim inf|an| = 0. Prove that there is a subsequence (ank ) ∞ k=1 such that P∞ k=1 ank converges. 5. For each of the following questions, prove that for any fixed x ∈ R, the series converges. Recall that x 0 = 1 for all x ∈ R by definition (in particular 00 = 1) and 0! = 1. (Hint: Treat the case x = 0 and x 6= 0 separately. For x 6= 0, use the ratio test) (a) X∞ n=0 x n n! (b) X∞ n=0 (−1)nx 2n+1 (2n + 1)! (c) X∞ n=0 (−1)nx 2n (2n)! Remark 0.1. These are the definitions of exp(x),sin(x) and cos(x) respectively.

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