c255exw97 - Final Examination Mathematics 189-255B 1...

Info icon This preview shows pages 1–2. Sign up to view the full content.

Final Examination April 30, 1997 Mathematics 189-255B 1. Define: (a) (i) the sequence of functions ( f n ) converges uniformly on S to a function f : S R ; (ii) the infinite series X n =1 f n converges uniformly on S . (b) Let f be a bounded function defined on [ a, b ] , ( -∞ < a < b < ). Define: (i) Upper (Darboux) sum U ( P ) of f with respect to the partition P of [ a, b ]; (ii) Upper and lower (Darboux) integrals Z b a fdx and Z b a fdx respectively; (iii) f is integrable on [ a, b ]. 2. (a) Prove the comparison test: Let X n =1 a n and X n =1 b n be two series of non negative terms. If a n b n for n N and X n =1 b n converges, so does X n =1 a n ; if the series X n =1 a n is divergent, so does X n =1 b n . (b) Show that X n =1 1 n log 1 + 1 n is convergent. 3. (a) Let X n =1 a n be a convergent series. If 0 < b n +1 b n for n N , prove that X n =1 a n b n is convergent. (b) Suppose that X n =1 a n n p is convergent. Show that X n =1 a n n q is convergent if q > p .
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern