1. Let an = 5n 4n3 5 This sequence converges to L = 4 (a) Find the smallest N EN such that |an — L| < 10-4 for all n > N. (b) Show that, for all n > 12, |an - L| < 1/n. Deduce an integer M (not necessarily the smallest!) such that an - L| < 10-6 for all n > M. 1| < 10 101 a 5 (c) Prove from first principles that an converges to .

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Let {an be the sequence Use MCT to show that {an}1 converges. an+1 = a₁ = 2, 1 3 an for n ≥ 2.

3. Prove the root test: Let {an} be a sequence of real numbers, and define p(n) = |a,|'/n. Prove the following: (i) If there exists c 1 so that p(n) N, then Ean converges absolutely. (ii) If for all N there exists n > N so that p(n) > 1, then an diverges.