UCLA MATH 131A - quiz5 (4 pages)

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Math 131A Real Analysis Quiz 5 Instructions You have 30 minutes to complete the quiz There are two problems worth a total of 24 points You may not use any books or notes Partial credit will be given for progress toward correct proofs Write your solutions in the space below the questions If you need more space use the back of the page Do not forget to write your name and UID in the space below Name Student ID number Question Points Score 1 8 2 8 3 8 Total 24 Problem 1 8pts Suppose f R R is differentiable and that f 0 R R is bounded Prove that f is uniformly continuous Solution Let 0 Since f 0 is bounded there exists an M 0 such that x R f 0 x M Let M suppose x y R and that x y If x y we have f x f y 0 Otherwise the MVT gives a z between x y and y such that f 0 z f x f and we see that x y f x f y f 0 z x y M x y M Problem 2 P a 4pts Suppose k 1 ak is a series P Define the sequence of partial sums sn n 1 of k 1 ak P b 4pts Suppose k 1 ak is a series with sequence of partial sums given by 1 1 n 2 n 1 Does the series P k 1 ak converge Why Solution P a sn nk 1 ak b Yes because the sequence of partial sums converges Problem 3 8pts P P Suppose that k 1 ak and k 1 bk are two series consisting of positive non zero terms P Suppose that limn abnn 0 and that k 1 bk is convergent P Prove that k 1 ak is convergent Help You should use the comparison test Solution Since limn an bn 0 we can find an N N so that an 1 bn i e n N n N an bn P P Since k 1 bk converges k N 1 bk converges P P By the comparison theorem k N 1 ak converges and so k 1 ak converges n N n N