澳大利亚数学Midterm代考 Math 327代写

Math 327 Fall 2016 Midterm 1

澳大利亚数学Midterm代考 You can use elementary algebra and any result that we proved in class (but not in the homework). You need to prove everything else.

Write clearly and legibly. Justify all your answers.

You will be graded for correctness and clarity of your solutions.

You may use one 8.5 x 11 sheet of notes; writing is allowed on both sides.

You can use elementary algebra and any result that we proved in class (but not in the homework). You need to prove everything else.

Please raise your hand and ask a question if anything is not clear. This exam contains 6 pages and is worth a total of 50 points.

You have 50 minutes. Good luck

NAME:

PROBLEM 1 (10 points)

PROBLEM 2 (17 points)

PROBLEM 3 (11 points)

PROBLEM 4 (12 points)

Total

Problem 1 (10 points) Find lim_n_→∞2n+1 and prove your result. 澳大利亚数学Midterm代考

Problem 2 Given S = { ₂_n₊₁|n ∈ N } Let s = sup S and i = inf S 澳大利亚数学Midterm代考

a)(6 points) Find the value of i and prove i = infS.

b)(6 points) Find the value of s and prove s = supS 澳大利亚数学Midterm代考

c)(3 points) Is S closed ? Justify your answer.

d)(2 points) Is S sequentially compact ? Justify your answer.

Problem 3(7 points)Prove that if the sequence {a_n} converges to a then the sequence {|a_n|} converges to |a|.(you can use the inequality (|x − y|≥||x| − |y||)

(4 points) Is it true that if a_n converges to a then a_n converges to a ? Justify your answer.

Problem 4(12 points) Say if each of the statements below is True or False (just write T or F next to each of them), and briefly explain why. 澳大利亚数学Midterm代考

1.A convergent sequence must be monotone and bounded.

Assume {a_n} converges , and {b_n} is a sequence.Then {a_n b_n} converges if and only if {b_n} converges.