MAT 137Y: Calculus! Problem Set 7. Due in tutorial on July 27/28 Instructions: Print this cover page, fill it out entirely, sign at the bottom, and STAPLE it to the front of your problem set (You do not need to print out the questions.) Doing this correctly is worth 1 mark. You need to turn in this assignment in the tutorial in which you are enrolled. Before you attempt this problem set, read sections 11.1 to 11.4, 12.2 and 12.3. Read \"A note on collaboration\" in page 5 of the course outline. Last name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . First name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Student number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tutorial Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TA's name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IMPORTANT NOTES ON COLLABORATION Solving a mathematical problem set has two parts: 1. The discovery phase. This is the time you spent trying to figure out how to solve the problems. You are welcome and encouraged to collaborate with other students in this phase as discussing problems with your classmates is a useful and mathematically healthy practice. 2. The Write-up phase. When it comes time to write up your solutions for submission, you must work independently and present solutions in your own words. Be alone when you write your solutions. If you collaborate on this part, or you copy part of your solutions from somebody else, or you have notes written by somebody else in front of you when you write your solutions, or you use a draft or sketch that you wrote in collaboration with somebody else, you are engaging in academic misconduct. The University of Toronto takes academic integrity very seriously. We are obligated to report all suspected instances of misconduct to OSAI. Please do not force us to do so. Every year, multiple students in this course are disciplined. For more information, see http://www.artsci.utoronto.ca/newstudents/transition/academic/plagiarism Please sign below to verify that you have read and understood the instructions on collaboration. Signature Date 1. Let f (x) be a continuous, positive and decreasing function. (a) Use the definition of limit to prove that Z Z f (x)dx converges the sequence an = n f (x)dx converges. 1 1 (b) Find a function f (f does not have to be decreasing or positive) such that the sequence Z n f (x)dx an = 1 converges but Z f (x)dx 1 diverges. 2. Consider the sequence ak = 1 . k Let ( bk = 1 ak + 1 if ak > 16 ak otherwise. 2 Find limk bk . 3. Let Z an = 0 1/n 2x dx. 3x Is an convergent. If it is, what does it converge to? Is an monotonic? If it is, classify it. 4. Let an be the sequence a1 , a2 , . . . , an , . . . . Let nk be a increasing sequence of natural numbers: 1 < n1 < n2 < < nk < . . . . We consider the sequence bk = ank . This sequence looks like b1 , b2 , b3 , . . . = an1 , an2 , . . . , ank , . . . . For example, if an = n1 and n1 = 3, n2 = 6, n3 = 9, etc. Then {bk } = {ank } = { 13 , 61 , 19 , }. The sequence ank called a subsequence of an . The sequence ank contains elements of the sequence an but we remove the elements an if n is not in the sequence nk . \u0001 (a) Prove that an = sin 4 n has a convergent subsequence. This shows that a divergence sequence can have a subsequence that is convergent. The next part of this question shows that the opposite cannot occur. (b) Prove that if bk = ank is a subsequence of an , then if an converges, so does bk and if an , then bk . 5. Determine whether the following sequences converge or diverge: 1 sin( k) ; k 4 2n (b) an = n ; n (a) an = 1 x(n+ n ) (c) an = for a fixed x [1, ) (hint: use special limits in section 11.4). n! 6. Determine whether the following series' converge or diverge: (a) (b) (c) (d) P k=1 P k=1 P k=4 P k=1 1 3+ln(k)+k 1 k2 (ln(k)+1)2 k4 1 k3 ln(k)1 k! (2k)