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Quiz 7 - Math 141 (7380) Instructions: • The quiz is worth 100 points. There are 5 problems, each worth 20 points. Your score on the quiz will be converted to a percentage and posted in your assignment folder with comments. • This quiz is open book and open notes, and you may take as long as you like on it provided that you submit the quiz no later than the due date posted in our course schedule of the syllabus. You may refer to your textbook, notes, and online classroom materials, but you may not consult anyone. • You must show all of your work to receive full credit. If a problem does not seem to require work, write a sentence or two to justify your answer. • Please write neatly. Illegible answers will be assumed to be incorrect. • Please remember to show ALL of your work on every problem. Read the basic rules for showing work below BEFORE you start working on the quiz: 1. Each step should show the complete expression or equation rather than a piece of it. 2. Each new step should follow logically from the previous step, following rules of algebra. 3. Each new step should be beneath the previous step. 4. The equal sign, =, should only connect equal numbers or expressions. • This quiz is due at 11:59 PM (Eastern Time) on Saturday, December 10. *********************** At the end of your quiz you must include the following dated statement with your name typed in lieu of a signature. Without this signed statement you will receive a zero. I have completed this quiz myself, working independently and not consulting anyone except the instructor. I have neither given nor received help on this quiz. Name: Date: 1 Quiz 7 - Convergence Tests of Series 1. Using the alternating test, determine whether ∞∑ n=0 sin ( 2n+1 2 π ) 1 + √ n is convergent or diver- gent. 2. Chapter 10-7, Problem 36. 3. Chapter 10-8, Problem 20. 4. Chapter 10-9, Problem 38. 5. Chapter 10-10, Problem 32. 2 xX x x ® k. xT x x x sin) =x 31 +37 -I7 +.= 3EDEFETT. = Lex IT 430 437 + =) Find a power series for sin( x?) 34. Find a power series for sin( 2x ) . Find a power series for ¢™ 36. Find a power series for ¢™*) Find a power series for cos( x ) 38. Find a power series for cos( x?) 1 Represent the integral as a merical series: J sin) ax 0 Represent the integral as a numeric series; Jo 0 n(x) a. fim SH-X x 0 x 109 Representing Functions as Power Series. Contemporary Calculus Practice Answers Practie 1: meric Sess Font 540 nit ey 1 Pa (2) Replacing "x" with "x> we have he 3 for IX 1<1 or=""><1 16.="" foxe”="" 21="" 5="" nlx"="" 5="" nlx="" +2)"="" n="l" n="1" n="0" in="" problems="" 25="" —="" 34,="" the="" letters="" "a"="" and="" "b"="" represent="" positive="" constants.="" find="" all="" values="" of="" x="" for="" which="" each="" given="" power="" series="" converges.="x-a)" 27.="" n="" n="1" 31="" 3="" (ax="b)"" 32="" 3="" (ax="" +b)”="" n="1" n="1" 108="" power="" series="" contemporary="" calculus="" 22.="" j="" cos(x")="" dx="" and="" j="" cos(x”)="" dx="" sin(x-)dx="" and="" j="" sin(x)="" dx="" ud="" dam="" i="" ha="" j="" ama="" sa="" 26="" j="" moa="" aa="" j="" moa="" 2d="" incom="" ana="" j="" fano="" in="" problems="" 28="" —="" 35,="" use="" the="" series="" representation="" of="" these="" functions="" 10="" calculate="" the="" limits.="" 0.="" jm="" 100="" xs0="" x1="" 3="" 0="" x0="" x="" x-sin(x)="" lim="" 0="" x="" 1="" use="" the="" series="" for="" ¢*="" and="" ¢™*="" 10="" write="" a="" series="" representation="" for="" cosh(x)="7" (&"="" +¢="" (the="" function="" "cosh"="" is="" called="" the="" hyperbolic="" cosine="" function.)="" use="" the="" series="" for="" ¢*="" and="" ¢™*="" to="" write="" a="" series="" representation="" for="" sinh(x)="" (the="" function="" "sinh"="" is="" called="" the="" hyperbolic="" sine="" function.)="" 10.10="" taylor="" and="" maclaurin="" series="" “ontemporary="" c="" 38.="" show="" that="" d(="" series="" for="" cosh(="" x="" )="" in="" problem="" 36="" )="" is="" the="" series="" for="" sinh(="" x="" )="" in="" problem="" 37.="" “the="" ratio="" test="" is="" commonly="" used="" with="" series="" that="" contain="" factorials,="" and="" factorials="" are="" also="" going="" 10="" become.="" more="" common="" in="" the="" next="" few="" sections.="" problems="" 31-40="" ask="" you="" to="" simplify="" factorial="" expressions="" in="" order="" to="" get="" ready="" for="" that="" usage.="" (definitions:="" 0!="1,11" =="" 1,2!="11222," 31="12:3" =="" 6,4!="24," and,="" in="" general,="" n!="12:3" ..="" {(n-1)'n="" )="" a="" 1="" :="" 1="" 31.="" show="" that="" tt="mT" .="" show="" that="" g3="ITE)" n!="" 1="" .="" (1="" 33.="" show="" that="" tu3)t="ETE" de="" 34.="" show="" that="" g="" a1!="" 1="" 35.="" show="" that="" {ni="" t)t="(sD)" (+1)="" (042)"..2n)="" (20)!="" qn="" 1="" 1="" 37.="" show="" that="" oh="mT" -="" show="" that="" t3(py="" dy)="Tanned)" 107="" absolute="" convergence="" and="" the="" ratio="" test="" “="" calculus="" ”="" 39.="" show="" that="" jr="a" 40.="" forn="">0, which is larger, 7! or 241% 5 o nil In problems 41 - 56, (a) determine the value of L= lim | <= |="" for="" the="" given="" series,="" and="" nx="" fr="">