DOC PREVIEW
UCLA MATH 131A - quiz5

This preview shows page 1 out of 4 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 4 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 4 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

Math 131AReal AnalysisQuiz 5Instructions: You have 30 minutes to complete the quiz. There are two problems, worth atotal of 24 points. You may not use any books or notes. Partial credit will be given for progresstoward correct proofs. Write your solutions in the space below the questions. If you need morespace 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 Score1 82 83 8Total: 24Problem 1. 8pts.Suppose f : R → R is differentiable and that f0: R → R is bounded.Prove that f is uniformly continuous.Solution: Let  > 0.Since f0is bounded there exists an M > 0 such that∀x ∈ R, |f0(x)| ≤ M.Let δ =M, suppose x, y ∈ R and that |x − y| < δ.If x = y we have |f(x) − f(y)| = 0 < . Otherwise, the MVT gives a z between xand y such that f0(z) =f(x)−f(y)x−yand we see that|f(x) − f(y)| = |f0(z)||x − y| ≤ M|x − y| < Mδ = .Problem 2.(a) [4pts.] SupposeP∞k=1akis a series.Define the sequence of partial sums (sn)∞n=1ofP∞k=1ak.(b) [4pts.] SupposeP∞k=1akis a series with sequence of partial sums given by1 −12n∞n=1.Does the seriesP∞k=1akconverge? Why?Solution:(a) sn=Pnk=1ak.(b) Yes, because the sequence of partial sums converges.Problem 3. 8pts.Suppose thatP∞k=1akandP∞k=1bkare two series consisting of positive non-zero terms.Suppose that limn→∞anbn= 0 and thatP∞k=1bkis convergent.Prove thatP∞k=1akis convergent.Help! You should use the comparison test.Solution: Since limn→∞anbn= 0, we can find an N ∈ N so that∀n ∈ N, n > N =⇒anbn< 1,i.e, ∀n ∈ N, n > N =⇒ |an| < |bn|.SinceP∞k=1bkconverges,P∞k=N +1bkconverges.By the comparison theoremP∞k=N +1akconverges, and


View Full Document

UCLA MATH 131A - quiz5

Download quiz5
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view quiz5 and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view quiz5 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?