Assignment 1 PDF

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Assignment 1...

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School of Mathematical Sciences Monash University Semester 2 - 2018

MTH1030 Assignment 1 Due: 16th of August 2018, 4 p.m.

(assignment box ground floor 9 Rainforest Walk)

The Rules of the Game For the most part these are pretty tough Monash Faculty of Science rules that apply to all assignments in maths units at uni. • The assignment must be submitted in hardcopy form in the assignment box (ground floor 9 Rainforest Walk) of your support class on Thursday the 16th of August 2018, 4 p.m. • If you do not submit your assignment in the correct assignment box, your assignment will not be marked and you will receive 0 marks on your assignment! • The maximum number of marks for this assignment is 60. This comprises 15 marks for the first, second and third set of questions, and 15 marks for presentation and communication. How exactly marks are split up is specified in the questions. To get full marks your report 1. must be neatly presented, but does not necessarily have to be typed; 2. must not contain any mistakes (mathematical, logical, spelling, etc.); 3. must be self-contained, including the statement of a problems at the beginning, and answers to whatever questions were asked at the end; 4. must be understandable without too much effort by other students taking this course; 5. must contain a clear description of the mathematics that justifies everything you do in full English sentences. A lot of this is about the way you present your work. Just to put a figure to it, you can lose up to 25% of the total marks for poor presentation. In particular, if your handwriting is illegible, whatever you are writing about will not be counted. If you have problems in this respect you may want to consider typing everything up after all. • Making things easier for yourself. To facilitate some of the more complicated and repetitive calculations in our assignments you should use Mathematica. As a Monash student you have access to this very powerful computational maths program. To download and register a free copy of Mathematica for use at home follow the instructions on our website. You can even simultaneously do all the calculations and write up your report nicely formatted using this program. Check out the sample Mathematica notebook and accompanying YouTube video on our Moodle page. • Penalties for late submission. The penalty for assignments submitted late is -10% per day late or part thereof. Weekends and holidays attract the same penalty as weekdays. No assignment can be accepted for assessment more than eight days after the due date except in exceptional circumstances and in consultation with Dr Norm Do. Late assignments can be submitted to Dr Santiago Barrera Acevedo. • Submit your own work. Please make sure to submit your own work. It is okay, and we would even encourage you to discuss a problem with other students who are taking the course to try to understand how to attack it (and to avoid going completely off the mark). However, in the end you have to do your own calculations and you have to write up your own report in your own words. Monash University is very picky about this last point and, to make sure that you understand how serious this is, requires that you attach a completed and signed Assessment Cover Sheet to the front of your assignment. Make sure that you understand what you are signing there. You can download a copy of the coversheet from the unit’s website. 1

Question 1 (15 marks) a) [1 mark] Explain exactly what it means for {an }n∈N to converge to L ∈ R. b) [1 mark] Suppose that a sequence is increasing and bounded above by some real number. Does the sequence have a limit? Justify your answer. c) [1 mark] Explain exactly what the partial sums of a series are. P∞ d) [1 mark] Explain what it means for a series n=0 an to converge to S ∈ R. P∞ e) [3 marks] The n-th term test states that if lim an 6= 0 then the infinite series n=0 an diverges. If n→∞ P∞ lim an = 0 does n=0 an converge? Give examples. n→∞

f) [8 marks] For the following series determine with explanation whether they converge or diverge. If possible also determine the limits for those among the sequences that converge.   ∞  ∞ ∞ ∞  X X X X 1 1 1 3 −n e (iii) (iv) (i) + (ii) πn 2n n n(n + 3) n=0 n=0 n=1 n=1

Question 2 (15 marks) a) [1 mark] Write down explicitly the ratio test for series. P∞ xn−1 b) [3 marks] Find the radius and interval of convergence of the power series n=1 n3n . ′

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c) [1 mark] Let f and f , f , . . . , f (n) be continuous in a closed interval [a, b] containing a point c. Write down the n-th Taylor polynomial of f (x) around x = c. 1 and its interval of convergence. d) [5 marks] Find the Maclaurin series (explicitly) of f (x) = 2−x

e) [5 marks] Let c be the last non-zero digit of your student ID number. Use Mathematica to plot the 1 and the first five partial sums of the Maclaurin series of f (x). function f (x) = c−x Question 3 (15 marks) The Indian mathematician Srinivasa Ramanujan1 discovered the identity ∞ √ X 1 8(4n)!(1103 + 26390n) = . π n=0 9801(n!)4 3964n Note that the series starts with n = 0. This means that in the following it makes sense to talk about the 0th term and the 0th partial sum of this series. This series converges at an amazing speed and was used in 1985 to compute the first 17,526,100 digits of π, which was the world record at the time. To prove this identity is quite tricky, so let’s do some things with it that are within our reach. a) [3 marks] Calculate the 0th term of Ramanujan’s series to approximate π. How many correct digits do you get (counting the 3 at the start as the first digit)? Calculate the 1st partial sum by adding the 0th and the 1st term of the series to approximate π. How many correct digits do you get? 2 1

This is the mathematician that the recent movie “The man who knew infinity” is all about. To coax Mathematica into displaying the first 40 digits of some number after the decimal point use the command N[number, 40]. For example, to display 40 digits of π use N[Pi, 40]. 2

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b) [3 marks] Using the ratio test, verify that the series is convergent. Do this by hand and show your work. Don’t just plug this into Mathematica. c) [3 marks] Let an denote the nth term of this series, starting with the 0th term a0 . It turns out (you don’t have to show this) that an+1 < Lan for n ≥ 1, where L is the limit that you calculated under b). Show that this implies that an < Ln−1 a1 for n ≥ 2.

d) [3 marks] Let Sn be the nth partial sum of our series starting with the 0th partial sum S0 = a0 . Show that (for n ≥ 1) 1 a Ln . 0 < − Sn < 1 1−L π e) [3 marks] Use this estimate to find as small an n as possible such that the first 17,526,100 digits of Sn coincide with those of 1 . π (Hint: Trying to formally solve the equation you get at this point can be tricky. Instead just try to find the correct n by trial and error.) Divide 17,526,100 by your number n to figure out how many additional correct digits of 1/π you get on average by adding one more term of the series. Remarks: Note that the reciprocal of an approximation of 1/π that shares 17,526,100 leading digits with 1/π will not necessarily have 17,526,100 leading digits in common with π. A little bit more work is needed to be able to guarantee that. To pose this problem I’ve been using very formal language. Just in case you are not used to this and just cannot parse a statement like an+1 < Lan for n ≥ 1 yet, try to translate what this means into plain English or ask your lecturer, one of your friends or your support class teacher to help you in this respect. In the case of an+1 < Lan a translation runs along these lines: “(Since L will be really small) given any term of the series the term following it will be at least by a factor L smaller. Anyway, if you have trouble in this respect please make sure to ask for help early on.

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