Telescoping Series Practice Problems With Answers

The Math Forum's Internet Math Library is a comprehensive catalog of Web sites and Web pages relating to the study of mathematics. Instructor: Math 10560, Worksheet, Series March 14, 2016 For realistic exam practice solve these problems without looking at your book and without using a calculator. Do not find a decimal approximation. You will learn. Free Online TELESCOPING SERIES Practice and Preparation Tests. To see this, do a limit comparison with the. A limit problem asks you to determine what the y value of a function is zeroing in on as the x value approaches a particular number. This calculator will find the sum of arithmetic, geometric, power, infinite, and binomial series, as well as the partial sum. After numerous tries I gave up. Determine all functions f : R Ñ R such that the equality f pr x s y q f p x qr f y qs. Not all series are telescoping. 417721 0321304349 1, or a geometric series, or a telescoping series? If so we are done and the series converges by our general theorems. Thanks for contributing an answer to Stack Overflow! Please be sure to answer the question. TELESCOPING SERIES Dosubsequent termscancel out previousterms in the sum? May have to use partial fractions, properties Problems 1-38 from Stewart’s Calculus. This sheet has three sections. Harold’s Series Convergence Tests Cheat Sheet 24 March 2016 1 Divergence or nth Term Test Series: Telescoping Series Test Series:. In practice, it pretty much works that way. Solve the initial-value problem y00 +8y0 +16y = 0, y(0) = 3, y0(0) = 6. Interested in Power Series Problems with answers - Integration Calculus Bookmark it to view later. 1 and parts of 10. Evaluate the following inde nite integral. A limit of 0 does not automatically mean that the series will not diverge. Radius of Convergence for a Power Series. Partial sums & series. X Worksheet by Kuta Software LLC. Divergence Test. A series that converges has a finite limit, that is a number that is approached. The students may find Questions 2 and 3 difficult. Solution: Let c k = tan 1 k. MyMathLab will tell you if your answer is correct or not and you can rework any problems you get wrong over and over until the due date and time for the problem set. This problem gives a kind of justification. I meant all the whole number unit fractions, and I just started to do it: 1 + 1/2. Data Structures Algorithms Questions and Answers has been designed with a special intention of helping students and professionals preparing for various Certification Exams and Job Interviews. For example, while the concept map activity in our study was for a unit on density, for training purposes it included an example concept map on a nonscience topic (roses), and a practice activity based on the water cycle. a) Sketch a graph of f(x) in the interval −3π < x < 3π b) Show that the Fourier series for f(x) in the interval −π < x < π is π 4 − 2 π cosx+ 1 32. Teach yourself calculus. There are 5 basic types of questions, and we will cover each of them in the following sections. For example, let n=−2. A series can have a sum only if the individual terms tend to zero. In addition, any auxilliary sequence will be symbolized as the sum, as n goes from 1 to infinity, of b[n]. I have bought from them for over 1 years and never had a problem with a return or damaged item. So I made a mock F=ma exam with a couple of my friends on AoPS. Augustin-Louis Cauchy eventually gave a rigorous definition of the sum of a (convergent) series, and for some time after this, divergent series were mostly excluded from mathematics. An easy way to figure out the values of a and b is to plug in some easy numbers. Enter each answer in it's exact form. This is not intended to be a practice version of the test. TELESCOPING SERIES Now let us investigate the telescoping series. In the speed test you are awarded 10 points for each correct answer and a point for each second that is left. Now apply Rule 1 to the first summation and Rule 2 to the second summation. com Use Coupon Code 5FREE to get the first five videos free 10 Problems - Sequences, Geometric Series, Telescoping Series and the Test for Div. Then each term of the series is multiplied by r (the common ratio of GP) and is written by shifting each term one step rightward, and then by subtracting rN from N to get (1 − r) N. Corequisite: MATH 133 (Linear algebra and geometry), or familiarity with vector geometry Restriction: Not open to students who have taken CEGEP course 201-303 or MATH 150, MATH 151 or MATH 227 Calendar description: Taylor series, Taylor's theorem in one and several variables. The material for Prelim 2 covers sections 8. 1, Section 6. MATH 102 Practice problems: estimates and power series 1. In practice, it pretty much works that way. More examples of evaluating series can be found on the following page: Series Convergence and Divergence Practice Examples 1; Series Convergence and Divergence Practice Examples 2; Series Convergence and Divergence Practice Examples 3; Series Convergence and Divergence Practice Examples 4; Series Convergence and Divergence Practice Examples 5. Up To 70% Off Everything Home! 2-Day Shipping. Telescoping series is a series where all terms cancel out except for the first and last one. Math 181, Final Exam, Spring 2012 Problem 1 Solution 1. P 1 n=1 tan n1 n3 Since tan is always less than ˇ 2, compare to ˇ 2n3 to show convergence. So I made a mock F=ma exam with a couple of my friends on AoPS. The database of problems is designed for my online and offline calculus tutoring program. The idea is that you give your answers as quickly as possible. I can direct you to the right examples for your problem quickly, as well as answer any of your questions about the problem and its explanation. For the "paper" problems, for each WebAssign problem, prepare a sketch of the region whose area you are finding. Probably 1 hour of work Due date: 9-19-18. Articles, problems, games and puzzles - in Algebra and many of which are accompanied by interactive Java illustrations and simulations. If it is convergent find the sum. After numerous tries I gave up. X∞ k=3 5 2k. Our mission is to provide a free, world-class education to anyone, anywhere. Infinite Series: A Compact Reference Compiled by Damon Scott Table 1: Basic Tests for Convergence Name When to use Hypotheses What you do What you conclude Geometric Series Test (a) You see a geometric series, one where each term is some fixed mul-tiple of the term before it. Search Result for telescoping. Back to TV Repair FAQ Table of Contents. This problem gives a kind of justification. n must be a positive integer. graphing the integrand to see where the problem lies. Select the tables you want to practice. Power Series SubDerivInteg practice for quiz. When the difference between each term and the next is a constant, it is called an arithmetic series. Sign up to join this community. o q bASl BlB Zr niVg8hnt osS 5r8ewsXenrZv Yecdj. Multiple choice questions should take about 4 minutes to complete. he's probably stared at a computer screen more than he's slept. I work through an example of proving that a series converges and finding the sum of the series using Partial Fractions to create a Telescoping Series. TELESCOPING SERIES Now let us investigate the telescoping series. Answers to August 2005 Regents Exam. 0 O qMcapd9e9 owFi9t Bh9 AIgn 7fXiGnLi8tTeZ sAsl fg 2e4bRrsa C Y2i. When I was young, I used to lie in bed and wonder what you would get if you added up all the fractions. Course outline. A limit of 0 does not automatically mean that the series will not diverge. Examples 11. First, note that the telescoping series method only works on certain fractions. ) Geometric Series. As always, feel free to ask if. Use the formula for the sum of a geometric series to determine the sum when a1=4 and r=2 and we have 12 terms. Length and content of the actual nal may vary, at the discretion of the instructor. Hints And Answers For Pulley Problems Hint and answer for Problem # 2. Look at the partial sums: because of cancellation of adjacent terms. We can then take the limit of the partial sum to see what the series converges to. Not all series are telescoping. Telescoping Series and Strategies for Testing Series. MyMathLab Answers for Midterm Practice Exam: The students who are learning math form the Pearson’s portal; have to give MyMathLab Midterm Practice Exams. This is an excerpt from my full length lesson. Math 76 Calculus II Practice MT2 Fall 2007 Solutions with Travis Kelm CSU Fresno 31 Oct 2007 - 11 AM Travis makes no guarantees regarding the accuracy of these solutions. The geometric series is used in the proof of Theorem 4. A series that diverges means either the partial sums have no limit or approach infinity. u Worksheet by Kuta Software LLC. Problem 6 (20 points) An antibiotic decays exponentially in the human body with a half-life of about 2. The series converges absolutely by the direct comparison test B. Special series: geometric series and p-series; telescoping series Instructions: These problems are for you to use to test yourself, after you have practiced with the routine homework assignments, to see how ready you are for Test 3. on or off camera, the affection is real, and jungkook can be awkward and meme-like, but there are times when it's too adorable because jungkook grudgingly reciprocates, like he's sacrificing his dignity and then a limb so that jimin knows for sure that he's not repulsed, just ungraceful when it comes to this stuff. We are trying to determine if the series 1/n+1 converges or diverges. To be able to do this, we will use the method of partial fractions to decompose the fraction that is common in some telescoping series. If any of the 6 elements is bent, then the whole structure is out of alignment. During quiz/review lessons I authorize collaboration, the use of textbooks, notes, and supplemental devices and material. to the answers you will see if you use your calculator, Wolfram Alpha, or the integral Comparison Test Practice Problems Recognize telescoping series. One hint to give them is that for a and b > 0, a < b ⇒ a2 < b2. ©a f2i0 g1t2 W OK yu 7t6a I kS 1o cf NtQwPa0rpei NLpL 0C S. This section provides a useful collection of sample Interview Questions and Multiple Choice Questions (MCQs) and their answers with appropriate explanations. A limit of 0 does not automatically mean that the series will not diverge. I called Classic Two Inch Series 4 Ft. The first term is a = 5. So the Limit Divergence Test alone cannot say anything more about series convergence or divergence here. Suggested Review Problems with full solution videos from CalcCoach. " For example:. Chapter Text. There will be basic questions about series and the n-th term test. While this idea is the one needed most frequently, our students sometimes fixate on the alternating series test, which is a very specialized test guaranteeing convergence. Telescoping Series Example. telescoping series; problems and applying your newly learned skills is the only way to keep all of these tests straight, so please be sure you are putting in the. 3 was replaced with telescoping series that are not geometric and that require re-indexing. Learning questions for Section S 1-4 Review questions: 1. Then the improper integrals of f and g with the same limits of integration behave the same way, ie either both converge or both diverge. I Therefore, we can conclude that the alternating series P 1 n=1 ( 1) n 1 converges. Suggested Review Problems with full solution videos from CalcCoach. 3 Geometric Series and Telescoping Series. It contain the topic Series Completion, Series Completion , Series. Since |r| < 1, the series is convergent. 12, which is known as the ratio test. It would require an understanding of the Outlook Object model as mentioned by @roryap or use of CDO to send an email directly through SMTP. Ignore the mass of the pulleys. Test each series for convergence. Advanced Problems in Mathematics is recommended as preparation for any undergraduate mathematics course, even for students who do not plan to take the Sixth Term Examination Paper. For the "paper" problems, for each WebAssign problem, prepare a sketch of the region whose area you are finding. Included are: Guided notes with completed examples on convergence and divergence of the Telescoping and Geometric Series Problem Set with. Subject : Harmony between lab & Clinical 'Practice 21. (Oliver Heaviside, quoted by Kline) In this chapter, we apply our results for sequences to series, or in nite sums. What is a convergent series?. the actual test. 1, Section 6. Select the most accurate statement about the series. Geometric series either get bigger or approach a single number. Corequisite: MATH 133 (Linear algebra and geometry), or familiarity with vector geometry Restriction: Not open to students who have taken CEGEP course 201-303 or MATH 150, MATH 151 or MATH 227 Calendar description: Taylor series, Taylor's theorem in one and several variables. These problems are intended to help you practice applying all the concepts and techniques we’ve learned so far. Help with the Telescoping Series. Solutions will be posted on the course webpage later, so you can use these to gauge your preparedness for the quiz. Examples 11. The series converges by the telescoping series test C. I have bought from them for over 1 years and never had a problem with a return or damaged item. ☀ Find Deals Electric Grills ☀ 18 Electric Grill With Versa Stand Base by Cuisinart Free Shipping On All Orders Over $49. It is one of the most commonly used tests for determining the convergence or divergence of series. Ticket Out the Door Power Series Day 1. Course outline. (a) Z 2 1 1 x2 +4 dx= 3ˇ 8 (b) Z 2 0 1 x 1 dx divergent (c) Z 1 0 ln p ydy= 1 2 (d) Z 1 0 x2e x3 dx= 1 3 (e) Z 3 0. The following rules apply: Total points available is 110; but the test is out of 100. I meant all the whole number unit fractions, and I just started to do it: 1 + 1/2. A series that diverges means either the partial sums have no limit or approach infinity. When I was young, I used to lie in bed and wonder what you would get if you added up all the fractions. Find the values of p for which the series is. The database of problems is designed for my online and offline calculus tutoring program. Integral Test to determine if a series converges or diverges. MyMathLab Answers for Midterm Practice Exam: The students who are learning math form the Pearson’s portal; have to give MyMathLab Midterm Practice Exams. Answer: The characteristic equation is r2 +8r +16 = 0, which factors as (r +4)2 = 0, so r = −4 is the only solution of the equation. P 1 n=1 tan n1 n3 Since tan is always less than ˇ 2, compare to ˇ 2n3 to show convergence. You get 10 seconds for each question. To learn more, see our tips on writing great. Telescoping Series and Strategies for Testing Series. in expanded form. 258 Chapter 11 Sequences and Series closer to a single value, but take on all values between −1 and 1 over and over. I will ask you to write up and hand in your work for some of the problems so. ) Since lim n!1 n! en = 1series O diverges, by the Test for Divergence. Geometric series either get bigger or approach a single number. “Math is not a spectator sport. An arithmetic sequence is an ordered series of numbers, in which the change in numbers is constant. Instructor: Math 10560, Worksheet, Series March 14, 2016 For realistic exam practice solve these problems without looking at your book and without using a calculator. Our mission is to provide a free, world-class education to anyone, anywhere. Subject : Harmony between lab & Clinical 'Practice 21. Multiple Choice Questions for Review In each case there is one correct answer (given at the end of the problem set). 6 (the term “telescoping refers to the nice cancellation below) The fact that we can just calculate by brute force is quite unusual in the big scheme of things however all the telescoping series work more or less like this example. You can keep the exam questions when you leave. They are not meant as a substitute for regular and diligent practice! Do the following problems as if you were. For example, while the concept map activity in our study was for a unit on density, for training purposes it included an example concept map on a nonscience topic (roses), and a practice activity based on the water cycle. Get an answer for 'Precalculus, Chapter 6, 6. Unless otherwise instructed, give exact answers, not approximations (e. Definition of Convergence and Divergence in Series The n th partial sum of the series a n is given by S n = a 1 + a 2 + a 3 + + a n. 1 SEQUENCES. The harmonic series is a special case of p series, with p=1. A series that converges has a finite limit, that is a number that is approached. Problem 6 (20 points) An antibiotic decays exponentially in the human body with a half-life of about 2. Fab'r Cologne Subject : Galvano Formed Copings 8 pre conference courses. We will now look at some more examples of evaluating telescoping series. Examples: P 1 n=0 1=2 n, P 1 n=0 ( n10) n=n!, P 1 n=0 n(x 1) , Be able to determine whether a series diverges, converges absolutely, or converges condition-. The series of interest will always by symbolized as the sum, as n goes from 1 to infinity, of a[n]. graphing the integrand to see where the problem lies. For example, let n=−2. Advanced Problems in Mathematics is recommended as preparation for any undergraduate mathematics course, even for students who do not plan to take the Sixth Term Examination Paper. Let N be the sum of the arithmetico geometric series. 5'' D Aluminum Contractor Series Mini Rolling Scaffolding With 250 Lb. Again, the trick here is to look at the partial sums and before we right out the sequence of partial sums. Ratio and root tests, rearrangement, Cauchy product formula. Examples range from easy to difficult and illustrate the in-depth presentation of theory. Let's Practice: An auditorium has 20 seats on the first row, 24 seats on the second row, 28 seats on the third row, and so on and has 30 rows of seats. 1, Problem 39' and find homework help for other Precalculus questions at eNotes College Algebra - stitz-zeager. Test for convergence or divergence. 1 - Enter the first term A1 in the sequence, the common ratio r and n n the number of terms in the sum then press enter. Chapter Text. The last problem is a bonus problem. (b) You see the base is fixed, and the expo-. Use MathJax to format equations. You get 10 seconds for each question. It is found by using one of the following formulas: Video lesson. 15 Problems - Convergence Tests and Absolute vs. Now apply Rule 1 to the first summation and Rule 2 to the second summation. N 7 iA ilelH RrSi hg Bhtwsh Qrqe ysMeVrPv 3eZdO. Solve the initial value problem y′ = x y4, y(0) = 2. Telescoping refers to the phenomenon when many terms in a large sum cancel out - so the sum "telescopes. Homework 6 posted. Exercise 2. Again, the trick here is to look at the partial sums and before we right out the sequence of partial sums. Math 114 Worksheet # 1: Integration by Parts 1. We explain calculus and give you hundreds of practice problems, all with complete, worked out, step-by-step solutions. Back to TV Repair FAQ Table of Contents. Calculate the partial sums of a series. u Worksheet by Kuta Software LLC. Then each term of the series is multiplied by r (the common ratio of GP) and is written by shifting each term one step rightward, and then by subtracting rN from N to get (1 − r) N. Rewrite 1 n2+3n+2 in the form a n+1+b n+2. Here, by r x s we denote the greatest integer not exceeding x. This sheet has three sections. therefore in a telescoping series Sbbnn 11 It follows that a telescoping series will converge if and only if bn approaches a finite number as n. Typically, students practice by working through lots of sample problems and checking their answers against those provided by the textbook or the instructor. For the "paper" problems, for each WebAssign problem, prepare a sketch of the region whose area you are finding. Convergence Tests for Infinite Series In this tutorial, we review some of the most common tests for the convergence of an infinite series $$ \sum_{k=0}^{\infty} a_k = a_0 + a_1 + a_2 + \cdots $$ The proofs or these tests are interesting, so we urge you to look them up in your calculus text. to the answers you will see if you use your calculator, Wolfram Alpha, or the integral Comparison Test Practice Problems Recognize telescoping series. on or off camera, the affection is real, and jungkook can be awkward and meme-like, but there are times when it's too adorable because jungkook grudgingly reciprocates, like he's sacrificing his dignity and then a limb so that jimin knows for sure that he's not repulsed, just ungraceful when it comes to this stuff. Determine whether the series $\sum_{n=1}^{\infty} \frac{1}{(2n - 1)(2n + 1)}$ is convergent or divergent. The following rules apply: Total points available is 110; but the test is out of 100. Make sure you can correctly answer questions involving telescoping series and partial sums. Conditional and absolute convergence. You should not therefore be discouraged if. ☀ Find Deals Electric Grills ☀ 18 Electric Grill With Versa Stand Base by Cuisinart Free Shipping On All Orders Over $49. In each part determine if the series is convergent or divergent. Further, areas adjacent at the point where the graph crosses from below to above are congruent and cancel (see the green shaded regions below). A more truthful answer would be to report the area as 300 m2; however, this format is somewhat misleading, since it could be interpreted to have three significant figures because of the zeroes. Series Divergent series are the devil, and it is a shame to base on them any demonstration whatsoever. To see this, do a limit comparison with the. telescoping series; problems and applying your newly learned skills is the only way to keep all of these tests straight, so please be sure you are putting in the. 17Calculus - You CAN ace calculus. To find the sum of an infinite series • k=1 ak we form the sequence of partial sums that are often denoted by Sn. Please note there will be a review session held tomorrow from 7:00pm to 9:00pm in 115 Van Hise by me answering questions and solving problems in my problem sets. Series Divergent series are the devil, and it is a shame to base on them any demonstration whatsoever. Math 76 Calculus II Practice MT2 Fall 2007 Solutions with Travis Kelm CSU Fresno 31 Oct 2007 - 11 AM Travis makes no guarantees regarding the accuracy of these solutions. x Advanced Problems in Mathematics I have given each problem a difficulty rating ranging from to. In general, whenever you want to know lim n→∞ f(n) you should first attempt to compute lim x→∞ f(x), since if the latter exists it is also equal to the first limit. Charges on capacitors in series are equal to each other and in this case also equal to the total charge. Your instructor might use some of these in class. (1) Evaluate the given integral or determine that it is divergent. SECOND SAMPLE FINAL FOR MATH 2J This sample 2J nal exam is meant for practice only. In addition, any auxilliary sequence will be symbolized as the sum, as n goes from 1 to infinity, of b[n]. It would require an understanding of the Outlook Object model as mentioned by @roryap or use of CDO to send an email directly through SMTP. Corequisite: MATH 133 (Linear algebra and geometry), or familiarity with vector geometry Restriction: Not open to students who have taken CEGEP course 201-303 or MATH 150, MATH 151 or MATH 227 Calendar description: Taylor series, Taylor's theorem in one and several variables. nth term test for Divergence Direct Comparison Test Geometric Series Test Limit Comparison Test p-Series Test Alternating Series Test Telescoping Series Test Ratio Test Integral Test Root Test 4. Materials produced at the University of Colorado Boulder for second-semester calculus. The better way to report the number would be to use scientific notation: 3 ´ 10 2 m 2. Solution: Let c k = tan 1 k. As usual, this can be challenging and practice is the key here. Get an answer for 'Precalculus, Chapter 6, 6. The cross product. I can direct you to the right examples for your problem quickly, as well as answer any of your questions about the problem and its explanation. This is a telescoping series: The given problem is the. Answers to thematic essays and document-based questions represent accept-able responses. Then each term of the series is multiplied by r (the common ratio of GP) and is written by shifting each term one step rightward, and then by subtracting rN from N to get (1 − r) N. • You must show enough work to justify your answers. Chapter Text. Ticket Out the Door Power Series Day 1. If the numerators do not cancel out completely, then telescoping series will no longer work. Because the California microstamping law is impossible to comply with, no new Ruger® pistols (or any other manufacturer's, for that matter) have been added to the California Roster of Handguns Certified for Sale since the law became effective in 2013. With the aid of patterns, pupils determine values of r. Squeeze Theorem for Sequences. For example, let n=−2. Occassionally, you may also get questions that test harmonic progression (HP) - likely to find such a question in CAT than in the TANCET. TELESCOPING SERIES Now let us investigate the telescoping series. Everyone is welcome if you want to come. he's probably stared at a computer screen more than he's slept. You use the data/information at your own risk. ) Since lim n!1 n! en = 1series O diverges, by the Test for Divergence. ) Sorting sequences: Sort these sequences into four piles: sequences that converge to 0, sequences that converge to something other than 0,. Homework 6 posted. 5'' D Aluminum Contractor Series Mini Rolling Scaffolding With 250 Lb. The first term is a = 5. How to find the sum of a finite Arithmetic Series! s n = n(t 1 + t n)/2 To find the sum of a finite arithmetic series, you need to know three things. At this point, the armature losses can be found, but this is a separately excited machine and it is important to remember to account for power flow in the field circuit. For example, if the daily time series denotes inventory level, then converting from daily to weekly simply entails taking the inventory level at the end of each week. 0 O qMcapd9e9 owFi9t Bh9 AIgn 7fXiGnLi8tTeZ sAsl fg 2e4bRrsa C Y2i. "it's not about the track. It is one of the most commonly used tests for determining the convergence or divergence of series. In this portion we are going to look at a series that is called a telescoping series. Math 181, Final Exam, Spring 2012 Problem 1 Solution 1. For problems 4- 28 make sure you provide the graph of the parametric equation with orientation and also convert the parametric equation into a rectangular equation. A geometric sequence is a sequence such that any element after the first is obtained by multiplying the preceding element by a constant called the common ratio which is denoted by r. When you sum the sequence by putting a plus sign between each pair of terms, you turn the sequence into a geometric series. Unless otherwise instructed, give exact answers, not approximations (e. Learning to mathematically analyze circuits requires much study and practice. Voltages on both capacitors connected in parallel are the same. The Þrst section will remind you ab out tec hniques and form ulas that you should kno w. On Stuvia you will find the most extensive lecture summaries written by your fellow students. Sign up to join this community. Review of vector geometry. Occassionally, you may also get questions that test harmonic progression (HP) - likely to find such a question in CAT than in the TANCET. Course outline. The book's emphasis is on building problem-solving skills. I have bought from them for over 1 years and never had a problem with a return or damaged item. In this portion we are going to look at a series that is called a telescoping series. Are you thinking for GATE Coaching for GATE 2020 Exam just call at Eii for best GATE Coaching Result. Free practice questions for Calculus 2 - Types of Series. Specifically, its partial sums are s N = 1 p 2 1 + 1 p 3 1 p 2 + + 1 p N+ 1 1 p N = 1 + 1 p N+ 1; which have limit lim N!1 s N = 1. Moreover, if the series converges, its sum is 11lim n n Sb b The series in (c) is a telescoping series of the form. Math 172 EXAM 2 Review problems Remember to bring photo ID to the test. SECOND SAMPLE FINAL FOR MATH 2J This sample 2J nal exam is meant for practice only. Partial sums & series. Infinite series can be very useful for computation and problem solving but it is often one of the most difficult. Always helpful customer service waiting to help in any way they can. Candidates choose 6 questions to answer. Arithmetic Series. Engineers Institute of India is Top Ranked GATE Coaching Institute with Highest Results. The second giv es a n um b er of practice questions for you to w or k on. Articles, problems, games and puzzles - in Algebra and many of which are accompanied by interactive Java illustrations and simulations. In practice, it pretty much works that way. 417721 0321304349 1, or a geometric series, or a telescoping series? If so we are done and the series converges by our general theorems. If the parts were perfectly rigid, then if any one is bent, all 6 are bent.