Analysis 1 (from week 9 on)
  • Official Unit description
  • Course materials:

    Lecture Notes (If you see any typos, etc., please, e-mail me.)

    REMARK ON ALTERNATING SERIES: the Notes do not really give the definition of the alternating series. There is only Thm 6.1.1 which says that if a positive sequence (a_n) decreases and converges to to zero, then the series a_1-a_2+a_3-a_4+... converges. The Theorem is called the "alternating series test". However, the alternating series is defined as any series, where terms come with alternating signs. Moreover, even though in the particular case of the decreasing sequence (a_n), the condition a_n->0 is sufficient for converginece of the alternating series, in general, it is NOT sufficient. Indeed: take the sum 1/2 - 1/4 + 1/3 - 1/6 + 1/4 - 1/8 +... = 1/2(1/2+1/3+1/4+...). As the sum has been written in the left-hand side: the signs alternate, a_n->0, but the series diverges, because the assumption that the sequence (a_n) decreases is not satisfied.

    Exercise sets (beginning form set 8, assigned on Firday of week 9, then from week 11 on): 8  9  10  11  12 13  14  15 

    Solutions:     8  9  10  11  12  13  14  15     

  • Weekly homework to do: (beginning with Exercises set 8, assigned on week 9)

    Week 9  Exercises 8: 2(b), 3(b,c,d), 5, 6, 7(a,d), 8(a,c,e).
    Week 11  Exercises 9: 1, 2, 3(a,b,d), 4. 
    Week 12  Exercises 9: 5; Exercises 10: 0, 1(a), 2(Theorem 1(i) and Theorem 2), 3. 
    Week 13 Exercises 10: 4; Exercises 11: 1, 2(c), 3, 4, 5(b,c), 6, 7.  
    Week 14 Exercises 12: 2-5, 8-10.    
    Week 15: Exercises 13: 1-6.    
    Week 16: Exercises 13: 7,9-12. Exercises 14: 1,2.    
    Week 17: Exercises 14: 3-5.    
    Week 18: Exercises 14: 6,7; Exercises 15: 2-4.    REMARK: Question 2 requiers an additional assumption that he sequence (a_n) decreases.
    Week 19: Exercises 15: 5,6,8,9. In Problem 8 note, but do not do (e). 
    NO HOMEWORK OVER EASTER BUT TUTORIALS WILLBE HELD ON THE FIRST WEEK AFTER THE BREAK! 
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  • Recommended textbooks
    • C.W. Clark, Elementary Mathematical Analysis. Wadsworth Publishers of Canada, 1982.
    • G. H. Hardy, A course of Pure Mathematics. Cambridge University Press, 1908.
    • J. M. Howie, Real analysis. Springer-Verlag, 2001.
    • S. Krantz, Real Analysis and Foundations. Second Edition. Chapman and Hall/CRC, 2005.
    • I. Stewart and D. Tall, The Foundations of Mathematics. Oxford University Press, 1977.
  • Past examination papers are available to download or view on Blackboard.

    You can find them in "Courses/Math Teaching/Exams/Past Examinations".


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