University of California Advanced Calculus of One Real Variable Exam Practice

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DVANCED CALCULUS 1,Basic topology of the real line, numerical sequences and series, continuity, differentiability, Riemann integration, uniform convergence, power series.

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QUIZ 4. ADVANCED CALCULUS 1, FALL 2020
1. (5 pts) Suppose that the sequence {xn } is increasing and has a
convergent subsequence. Does {xn } necessarily have a limit? Prove
your answer.
2. (5 pts) Which three of the following statements, if combined,
imply that the sequence {x2n−1 }∞
n=1 converges? Prove your answer.
∞
(A) the sequence {xn }n=1 is bounded
(B) |xn − xn+100 | < n1 for every n ∈ N (C) the sequence {x2n }∞ n=1 is increasing 1 (D) |xn − xn+101 | < n for every n ∈ N 3. (5 pts) Use the Cauchy criterion to prove that the sequence converges 1 1 1 xn = 1 + 2 + 2 + ... + 2 . 2 3 n Hint: Use the inequality 1 1 1 < − . 2 n n−1 n 1 Purchase answer to see full attachment Tags: sequence converges Cauchy sequence convergent subsequence theorem limit limit behavior User generated content is uploaded by users for the purposes of learning and should be used following Studypool's honor code & terms of service.