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UTS MATH3068 Analysis Sol olny for questions 2,3,4




Question;The University of SydneySchool of Mathematics and StatisticsAssignment 1MATH3068 AnalysisSemester 2, 2014Web Page: Daniel HauerThis assignment is worth 5% of your assessment for this unit and is due on Friday August 29, before3 pm. Your written answers to this assignment should be stapled into a folder and handed in to theSchool of Mathematics and Statistics Student O?ce, Carslaw room 520. Your signature willbe required when you hand your assignment in.All necessary working must be shown. A completed cover sheet (available from the School?s TeachingProgram web page, left hand column) must be attached to the front of your assignment, inside the folder.No assignment will be marked without the cover sheet attached.1.(a) Use the formal de?nition of convergence of a sequence (the and N de?nition) to prove thatif an = 4n+2, n? 0, then (an) converges to 4 as n??.n+1(b) For the sequence in part (a),?nd an integer N such that whenever n > N, the di?erence|an? 4| is less than 10?4.2. Establish the convergence or divergence of each of the following sequences (an), n? 1, showingworking. Where a sequence is shown to converge,?nd its limit.(a) an =(c) an =(b) an =(9n2? 3n)? 3n,2n?(?2)n,nn(d) an =n!e2nnn,1?2+3?4+......+(2n?1)?2n?.n2 +13. If z = 3? i and w = 1 + 3i,?nd(a) z + w(b) z? w(c) |z|(d) zwzw(e)4. Test the following series for convergence or divergence, showing working.(a)?(?1)n?n=1 ln(n+1),(b)?n=1?(1 + n2? n),(c)?1+cos nn=1 3+n2,(d)?n=1lnnn+1


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