MATH 210 Exam 1, Fall 2013

Math 210 Quiz # 1, October 12, 2013 1. (a) Define each of the following: limit point, open set, compact set, bounded set. (b) Prove that a closed subset of a compact set is compact. (c) If E ⊂ K and K is compact, prove that E ⊂ K. 2. State without proof the completeness property of R, the Archimedean property of R, the Heine-Borel Theorem, and the Bolzano-Weierstrass Theorem ( for sequences). 3. (a) Use the completeness property of R to prove that every monotone decreasing sequence which is bounded below is convergent. (b) Let cn be the sequence defined by cn = 2 1 + 12 + 2.2 1 + 22 + · · · + 2n 1 + n2 − ln(1 + n2). Prove that cn is a convergent sequence. 4. A sequence an is defined by an+1 = an + n (1 + n2)an , a1 = 1. Obtain the asymptotic behaviour of an and prove your answer. 5. Given that �∞ k=1 bk is a convergent series of positive terms. Suppose that {ak} is a sequence satisfying the inequality |ak+1 − ak| ≤ ck, k ∈ N. Prove that ak is a convergent sequence. 6. Let �∞ k=1 ak be a series of positive terms, and use the notation An = n � k=1 ak, A = ∞ � k=1 ak. Suppose that lim sup k→∞ ak+1 ak = α, that α < 1, and take r such that α < r < 1. (a) Prove that the series is convergent. (b) Prove that there exists N > 0 such that 0 ≤ A − An ≤ aNrn−N+1 1 − r , n > N. 1