DOCSLIB.ORG

The Direct Comparison Test (Day #1)

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Direct Comparison Test (Day #1) 9.4--The Direct Comparison Test (day #1) When applying the Direct Comparison Test, you will usually compare a tricky series against a geometric series or a p-series whose convergence/divergence is already known. Use the Direct Comparison Test to determine the convergence or divergence of the series: ∞ 1) 1 3 n + 7 n=1 9.4--The Direct Comparison Test (day #1) Use the Direct Comparison Test to determine the convergence or divergence of the series: ∞ 2) 1 n - 2 n=3 Use the Direct Comparison Test to determine the convergence or divergence of the series: ∞ 3) 5n n 2 - 1 n=1 9.4--The Direct Comparison Test (day #1) Use the Direct Comparison Test to determine the convergence or divergence of the series: ∞ 4) 1 n 4 + 3 n=1 9.4--The Limit Comparison Test (day #2) The Limit Comparison Test is useful when the Direct Comparison Test fails, or when the series can't be easily compared with a geometric series or a p-series. 9.4--The Limit Comparison Test (day #2) Use the Limit Comparison Test to determine the convergence or divergence of the series: 5) ∞ 3n2 - 2 n3 + 5 n=1 Use the Limit Comparison Test to determine the convergence or divergence of the series: 6) ∞ 4n5 + 9 10n7 n=1 9.4--The Limit Comparison Test (day #2) Use the Limit Comparison Test to determine the convergence or divergence of the series: 7) ∞ 1 n 4 - 1 n=1 Use the Limit Comparison Test to determine the convergence or divergence of the series: 8) ∞ 5 3 2 n - 6 n=1 9.4--Comparisons of Series (day #3) (This is actually a review of 9.1-9.4) page 630--Converging or diverging? Tell which test you used. ∞ 29) 3 n (C) p-Series Test n n=1 page 630--Converging or diverging? Tell which test you used. ∞ 30) n 7 (-1/7) (B) Geometric Series Test n=0 9.4--Comparisons of Series (day #3) (This is actually a review of 9.1-9.4) page 630--Converging or diverging? Tell which test you used. ∞ 31) 1 (F) Direct Comparison Test n 5 + 1 n=1 page 630--Converging or diverging? Tell which test you used. ∞ 32) 1 (F) Direct Comparison Test 3 n - 8 n=2 9.4--Comparisons of Series (day #3) (This is actually a review of 9.1-9.4) page 630--Converging or diverging? Tell which test you used. ∞ 33) 2n (A) n-th Term Test 3n - 2 n=1 page 630--Converging or diverging? Tell which test you used. ∞ 34) 1 1 - (D) Telescoping Series Test n + 1 n + 2 n=1 9.4--Comparisons of Series (day #3) (This is actually a review of 9.1-9.4) page 630--Converging or diverging? Tell which test you used. ∞ 35) n 2 2 (E) Integral Test (n + 1) n=1 page 630--Converging or diverging? Tell which test you used. ∞ 36) 3 (G) Limit Comparison Test n (n + 3) n=1 9.4--Comparisons of Series (day #3) page 631--Converging or diverging? How can you tell? 45) ∞ 1 200n n=1 46) ∞ 1 200+10n n=0 page 631--Converging or diverging? How can you tell? 47) ∞ 1 2 200 + n n=1 48) ∞ 1 3 200 + n n=1 .
Load more

Recommended publications
  • Series: Convergence and Divergence Comparison Tests
    Series: Convergence and Divergence Here is a compilation of what we have done so far (up to the end of October) in terms of convergence and divergence. • Series that we know about: P∞ n Geometric Series: A geometric series is a series of the form n=0 ar . The series converges if |r| < 1 and 1 a1 diverges otherwise . If |r| < 1, the sum of the entire series is 1−r where a is the first term of the series and r is the common ratio. P∞ 1 2 p-Series Test: The series n=1 np converges if p1 and diverges otherwise . P∞ • Nth Term Test for Divergence: If limn→∞ an 6= 0, then the series n=1 an diverges. Note: If limn→∞ an = 0 we know nothing. It is possible that the series converges but it is possible that the series diverges. Comparison Tests: P∞ • Direct Comparison Test: If a series n=1 an has all positive terms, and all of its terms are eventually bigger than those in a series that is known to be divergent, then it is also divergent. The reverse is also true–if all the terms are eventually smaller than those of some convergent series, then the series is convergent. P P P That is, if an, bn and cn are all series with positive terms and an ≤ bn ≤ cn for all n sufficiently large, then P P if cn converges, then bn does as well P P if an diverges, then bn does as well. (This is a good test to use with rational functions.
    [Show full text]
  • Calculus II Chapter 9 Review Name______
    Calculus II Chapter 9 Review Name___________ ________________________ SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence. an 1) a = 1, a = 1) 1 n+1 n + 2 Find a formula for the nth term of the sequence. 1 1 1 1 2) 1, - , , - , 2) 4 9 16 25 Find the limit of the sequence if it converges; otherwise indicate divergence. 9 + 8n 3) a = 3) n 9 + 5n 3 4) (-1)n 1 - 4) n Determine if the sequence is bounded. Δn 5) 5) 4n Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges. 3 3 3 3 6) 3 - + - + ... + (-1)n-1 + ... 6) 8 64 512 8n-1 7 7 7 7 7) + + + ... + + ... 7) 1·3 2·4 3·5 n(n + 2) Find the sum of the series. Q n 9 8) _ (-1) 8) 4n n=0 Use partial fractions to find the sum of the series. 3 9) 9) _ (4n - 1)(4n + 3) 1 Determine if the series converges or diverges; if the series converges, find its sum. 3n+1 10) _ 10) 7n-1 1 cos nΔ 11) _ 11) 7n Find the values of x for which the geometric series converges. n n 12) _ -3 x 12) Find the sum of the geometric series for those x for which the series converges.
    [Show full text]
  • Drainage Geocomposite Workshop
    January/February 2000 Volume 18, Number 1 Drainage geocomposite workshop By Gregory N. Richardson, Sam R. Allen, C. Joel Sprague A number of recent designer’s columns have focused on the design of drainage geocomposites (Richard- son and Zhao, 1998), so only areas of concern are discussed here. Two design considerations requiring the most consideration by a designer are: 1) assumed rate of fluid inflow draining into the geocomposite, and 2) long-term reduction and safety factors required to assure adequate performance over the service life of the drainage system. In regions of the United States that are not arid or semi-arid, a reasonable upper limit for inflow into a drainage layer can be estimated by assuming that the soil immediately above the drain layer is saturated. This saturation produces a unit gradient flow in the overlying soil such that the inflow rate, r, is approximately θ equal to the permeability of the soil, ksoil. The required transmissivity, , of the drainage layer is then equal to: θ = r*L/i = ksoil*L/i where L is the effective drainage length of the geocomposite and i is the flow gradient (head change/flow length). This represents an upper limit for flow into the geocomposite. In arid and semi-arid regions of the USA saturation of the soil layers may be an overly conservative assumption, however, no alternative rec- ommendations can currently be given. Regardless of the assumed r value, a geocomposite drainage layer must be designed such that flow is not in- advertently restricted prior to removal from the drain.
    [Show full text]
  • Summary of Tests for Series Convergence
    Calculus Maximus Notes 9: Convergence Summary Summary of Tests for Infinite Series Convergence Given a series an or an n1 n0 The following is a summary of the tests that we have learned to tell if the series converges or diverges. They are listed in the order that you should apply them, unless you spot it immediately, i.e. use the first one in the list that applies to the series you are trying to test, and if that doesn’t work, try again. Off you go, young Jedis. Use the Force. Remember, it is always with you, and it is mass times acceleration! nth-term test: (Test for Divergence only) If liman 0 , then the series is divergent. If liman 0 , then the series may converge or diverge, so n n you need to use a different test. Geometric Series Test: If the series has the form arn1 or arn , then the series converges if r 1 and diverges n1 n0 a otherwise. If the series converges, then it converges to 1 . 1 r Integral Test: In Prison, Dogs Curse: If an f() n is Positive, Decreasing, Continuous function, then an and n1 f() n dn either both converge or both diverge. 1 This test is best used when you can easily integrate an . Careful: If the Integral converges to a number, this is NOT the sum of the series. The series will be smaller than this number. We only know this it also converges, to what is anyone’s guess. The maximum error, Rn , for the sum using Sn will be 0 Rn f x dx n Page 1 of 4 Calculus Maximus Notes 9: Convergence Summary p-series test: 1 If the series has the form , then the series converges if p 1 and diverges otherwise.
    [Show full text]
  • MATH 162: Calculus II Differentiation
    MATH 162: Calculus II Framework for Mon., Jan. 29 Review of Differentiation and Integration Differentiation Definition of derivative f 0(x): f(x + h) − f(x) f(y) − f(x) lim or lim . h→0 h y→x y − x Differentiation rules: 1. Sum/Difference rule: If f, g are differentiable at x0, then 0 0 0 (f ± g) (x0) = f (x0) ± g (x0). 2. Product rule: If f, g are differentiable at x0, then 0 0 0 (fg) (x0) = f (x0)g(x0) + f(x0)g (x0). 3. Quotient rule: If f, g are differentiable at x0, and g(x0) 6= 0, then 0 0 0 f f (x0)g(x0) − f(x0)g (x0) (x0) = 2 . g [g(x0)] 4. Chain rule: If g is differentiable at x0, and f is differentiable at g(x0), then 0 0 0 (f ◦ g) (x0) = f (g(x0))g (x0). This rule may also be expressed as dy dy du = . dx x=x0 du u=u(x0) dx x=x0 Implicit differentiation is a consequence of the chain rule. For instance, if y is really dependent upon x (i.e., y = y(x)), and if u = y3, then d du du dy d (y3) = = = (y3)y0(x) = 3y2y0. dx dx dy dx dy Practice: Find d x d √ d , (x2 y), and [y cos(xy)]. dx y dx dx MATH 162—Framework for Mon., Jan. 29 Review of Differentiation and Integration Integration The definite integral • the area problem • Riemann sums • definition Fundamental Theorem of Calculus: R x I: Suppose f is continuous on [a, b].
    [Show full text]
  • Calculus 141 Section 9.6 Lecture Notes
    Calculus 141, section 9.6 Ratio Test and Root Tests notes by Tim Pilachowski ∞ c r m ● The geometric series ∑ c r n = if and only if r <1. n = m 1 − r ∞ 1 ● The p-series converges whenever p > 1 and diverges whenever 0 < p ≤ 1. ∑ p n =1 n ∞ ∞ ● The Integral Test states a series a converges if and only if f ()x dx converges. ∑ n ∫1 n =1 ∞ ● In the Direct Comparison Test, ∑ an converges if its terms are less than those of a known convergent n =1 series, and diverges if its terms are greater than those of a known convergent series. ∞ an ● The Limit Comparison Test states: If lim exists and is a positive number, then positive series ∑ an n → ∞ bn n =1 ∞ and ∑ bn either both converge or both diverge. n =1 A downside to the Comparison Tests is that they require a suitable series to use for the comparison. In contrast, the Ratio Test and the Root Test require only the series itself. ∞ ∞ an+1 Ratio Test (Theorem 9.15) Given a positive series an for which lim = r : a. If 0 ≤ r < 1, then an ∑ n→∞ ∑ n =1 an n =1 ∞ an+1 converges. b. If r > 1, then an diverges. c. If r = 1 or lim does not exist, no conclusion. ∑ n→∞ n =1 an The proof of part a. relies upon the definition of limits and the “creation” of a geometric series which converges to which we compare our original series. Briefly, by the nature of inequalities there exists a value s for which an+1 0 ≤ r < s < 1.
    [Show full text]
  • MATH 166 TEST 3 REVIEW SHEET General Comments and Advice
    MATH 166 TEST 3 REVIEW SHEET General Comments and Advice: The student should regard this review sheet only as a sample of potential test problems, and not an end-all-be-all guide to its content. Anything and everything which we have discussed in class is fair game for the test. The test will cover roughly Sections 10.1, 10.2, 10.3, 10.4, and 10.5. Don't limit your studying to this sheet; if you feel you don't fully understand a particular topic, then try to do more problems out of your textbook! Summary of Relative Growth Rates. For two sequences (an) and (bn) such that (an) ! 1 and (bn) ! 1, let the symbols (an) << (bn) denote that (bn) grows faster than (an). Then the following is a summary of our most commonly encountered growth rates: (ln n) << ((ln n)r) << (nq) << (n) << (np) << (bn) << (n!) << (nn), where r > 1, 0 < q < 1, p > 1, and b > 1. General Strategy for Determining Convergence of a Series. There is no hard and fast rule for determining whether a series converges or diverges, and in general the student should build up their own intuition on the subject through practice. However, the following mental checklist can be a helpful guideline. 1 X (1) Divergence Test. Given a series an, the first thing one should always n=1 ask is, do the terms (an) ! 0? If not, we are done and the series diverges by the DT. If the terms do shrink to zero, however, we have to put more thought into the problem.
    [Show full text]
  • Direct and Limit Comparison Tests
    Joe Foster Direct and Limit Comparison Tests We have seen that a given improper integral converges if its integrand is less than the integrand of another integral known to converge. Similarly, a given improper integral diverges if its integrand is greater than the integrand of another integral known to diverge. We now apply the same idea to infinite series instead. Direct Comparison Test for Series: If0 a b for all n N, for some N, then, ≤ n ≤ n ≥ ∞ ∞ 1. If bn converges, then so does an. nX=1 nX=1 ∞ ∞ 2. If an diverges, then so does bn. nX=1 nX=1 a The Limit Comparison Test: Suppose a > 0 and b > 0 for all n. If lim n = L, where L is finite and L> 0, then n n →∞ n bn the two series an and bn either both converge or both diverge. X X ∞ 1 Example 1: Determine whether the series converges or diverges. 2n + n nX=1 1 1 We have for all n 1. So, 2n + n ≤ 2n ≥ ∞ ∞ 1 1 . 2n + n ≤ 2n nX=1 nX=1 1 Since the series on the right is a geometric series with r = , it converges. So we determine that our series of interest also 2 converges. ∞ 1 Example 2: Determine whether the series converges or diverges. n2 + n +1 nX=1 1 1 We have for all n 1. So, n2 + n +1 ≤ n2 ≥ ∞ ∞ 1 1 . n2 + n +1 ≤ n2 nX=1 nX=1 Since the series on the right is a p series with p = 2 > 1, it converges.
    [Show full text]
  • Series Convergence Tests Math 122 Calculus III D Joyce, Fall 2012
    Series Convergence Tests Math 122 Calculus III D Joyce, Fall 2012 Some series converge, some diverge. Geometric series. We've already looked at these. We know when a geometric series 1 X converges and what it converges to. A geometric series arn converges when its ratio r lies n=0 a in the interval (−1; 1), and, when it does, it converges to the sum . 1 − r 1 X 1 The harmonic series. The standard harmonic series diverges to 1. Even though n n=1 1 1 its terms 1, 2 , 3 , . approach 0, the partial sums Sn approach infinity, so the series diverges. The main questions for a series. Question 1: given a series does it converge or diverge? Question 2: if it converges, what does it converge to? There are several tests that help with the first question and we'll look at those now. The term test. The only series that can converge are those whose terms approach 0. That 1 X is, if ak converges, then ak ! 0. k=1 Here's why. If the series converges, then the limit of the sequence of its partial sums n th X approaches the sum S, that is, Sn ! S where Sn is the n partial sum Sn = ak. Then k=1 lim an = lim (Sn − Sn−1) = lim Sn − lim Sn−1 = S − S = 0: n!1 n!1 n!1 n!1 The contrapositive of that statement gives a test which can tell us that some series diverge. Theorem 1 (The term test). If the terms of the series don't converge to 0, then the series diverges.
    [Show full text]
  • Comparison Tests for Integrals
    Joe Foster Direct and Limit Comparison Tests Tests for Convergence: When we cannot evaluate an improper integral directly, we try to determine whether it con- verges of diverges. If the integral diverges, we are done. If it converges we can use numerical methods to approximate its value. The principal tests for convergence or divergence are the Direct Comparison Test and the Limit Comparison Test. Direct Comparison Test for Integrals: If0 ≤ f(x) ≤ g(x) on the interval (a, ∞], where a ∈ R, then, ∞ ∞ 1. If g(x) dx converges, then so does f(x) dx. ˆa ˆa ∞ ∞ 2. If f(x) dx diverges, then so does g(x) dx. ˆa ˆa Why does this make sense? 1. If the area under the curve of g(x) is finite and f(x) is bounded above by g(x) (and below by 0), then the area under the curve of f(x) must be less than or equal to the area under the curve of g(x). A positive number less that a finite number is also finite. 2. If the area under the curve of f(x) is infinite and g(x) is bounded below by f(x), then the area under the curve of g(x) must be “less than or equal to” the area under the curve of g(x). Since there is no finite number “greater than” infinity, the area under g(x) must also be infinite. Example 1: Determine if the following integral is convergent or divergent. ∞ cos2(x) 2 dx. ˆ2 x cos2(x) cos2(x) We want to find a function g(x) such that for some a ∈ R, f(x)= ≤ g(x) or f(x)= ≥ g(x) for all x ≥ a.
    [Show full text]
  • Long Term Test of Buffer Material. Final Report on the Pilot Parcels
    SEO100048 Technical Report TR-00-22 Long term test of buffer material Final report on the pilot parcels Ola Karnland, Torbjorn Sanden, Lars-Erik Johannesson Clay Technology AB Trygve E Eriksen, Mats Jansson, Susanna Wold Royal Institute of Technology Karsten Pedersen, Mehrdad Motamedi Goteborg University Bo Rosborg Studsvik Material AB December 2000 Svensk Karnbranslehantering AB Swedish Nuclear Fuel and Waste Management Co Box 5864 102 40 Stockholm Tel 08-459 84 00 Fax 08-661 57 19 32/ 07 PLEASE BE AWARE THAT ALL OF THE MISSING PAGES IN THIS DOCUMENT WERE ORIGINALLY BLANK Long term test of buffer material Final report on the pilot parcels Ola Karnland, Torbjorn Sanden, Lars-Erik Johannesson Clay Technology AB Trygve E Eriksen, Mats Jansson, Susanna Wold Royal Institute of Technology Karsten Pedersen, Mehrdad Motamedi Goteborg University Bo Rosborg Studsvik Material AB December 2000 Keywords: bacteria, bentonite, buffer, clay, copper, corrosion, diffusion, field experiment, LOT, mineralogy, montmorillonite, physical properties, repository, Aspo. This report concerns a study, which was conducted for SKB. The conclusions and viewpoints presented in the report are those of the authors and do not necessarily coincide with those of the client. Abstract The "Long Term Test of Buffer Material" (LOT) series at the Aspo HRL aims at checking models and hypotheses for a bentonite buffer material under conditions similar to those in a KBS3 repository. The test series comprises seven test parcels, which are exposed to repository conditions for 1, 5 and 20 years. This report concerns the two completed pilot tests (1-year tests) with respect to construction, field data and laboratory results.
    [Show full text]
  • Math Flow Chart
    Updated November 12, 2014 MATH FLOW CHART Grade 6 Grade 7 Grade 8 Grade 9 Grade 10 Grade 11 Grade 12 Calculus Advanced Advanced Advanced Math AB or BC Algebra II Algebra I Geometry Analysis MAP GLA-Grade 8 MAP EOC-Algebra I MAP - ACT AP Statistics College Calculus Algebra/ Algebra I Geometry Algebra II AP Statistics MAP GLA-Grade 8 MAP EOC-Algebra I Trigonometry MAP - ACT Math 6 Pre-Algebra Statistics Extension Extension MAP GLA-Grade 6 MAP GLA-Grade 7 or or Algebra II Math 6 Pre-Algebra Geometry MAP EOC-Algebra I College Algebra/ MAP GLA-Grade 6 MAP GLA-Grade 7 MAP - ACT Foundations Trigonometry of Algebra Algebra I Algebra II Geometry MAP GLA-Grade 8 Concepts Statistics Concepts MAP EOC-Algebra I MAP - ACT Geometry Algebra II MAP EOC-Algebra I MAP - ACT Discovery Geometry Algebra Algebra I Algebra II Concepts Concepts MAP - ACT MAP EOC-Algebra I Additional Full Year Courses: General Math, Applied Technical Mathematics, and AP Computer Science. Calculus III is available for students who complete Calculus BC before grade 12. MAP GLA – Missouri Assessment Program Grade Level Assessment; MAP EOC – Missouri Assessment Program End-of-Course Exam Mathematics Objectives Calculus Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Linear and Nonlinear Functions • Write equations of linear and nonlinear functions in various formats.
    [Show full text]