DOCSLIB.ORG

11.3-11.4 Integral and Comparison Tests

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
11.3-11.4 Integral and Comparison Tests 11.3-11.4 Integral and Comparison Tests The Integral Test: Suppose a function f(x) is continuous, positive, and decreasing on [1; 1). Let an 1 P R 1 be defined by an = f(n). Then, the series an and the improper integral 1 f(x) dx either BOTH n=1 CONVERGE OR BOTH DIVERGE. Notes: • For the integral test, when we say that f must be decreasing, it is actually enough that f is EVENTUALLY ALWAYS DECREASING. In other words, as long as f is always decreasing after a certain point, the \decreasing" requirement is satisfied. • If the improper integral converges to a value A, this does NOT mean the sum of the series is A. Why? The integral of a function will give us all the area under a continuous curve, while the series is a sum of distinct, separate terms. • The index and interval do not always need to start with 1. Examples: Determine whether the following series converge or diverge. 1 n2 • X n2 + 9 n=1 1 2 • X n2 + 9 n=3 1 1 n • X n2 + 1 n=1 1 ln n • X n n=2 Z 1 1 p-series: We saw in Section 8.9 that the integral p dx converges if p > 1 and diverges if p ≤ 1. So, by 1 x 1 1 the Integral Test, the p-series X converges if p > 1 and diverges if p ≤ 1. np n=1 Notes: 1 1 • When p = 1, the series X is called the harmonic series. n n=1 • Any constant multiple of a convergent p-series is also convergent. Example: 1 2017 • X p n n n=1 2 1 1 P P Comparison Test: Suppose an and bn are two series with positive terms. n=1 n=1 1 1 P P • If an ≤ bn for all n and if bn converges, then an also converges. Just like with improper integrals, n=1 n=1 if the larger converges, so does the smaller. 1 1 P P • If an ≥ bn for all n and if bn diverges, then an also diverges. Just like with improper integrals, if n=1 n=1 the smaller diverges, so does the larger. Examples: Determine whether the following series converge or diverge. 1 n2 • X p 9 3 n=1 n + n + 10 1 6n + 1 • X n2 − n − 3 n=3 1 n3 − 2 • X 8n4 + 6n n=5 1 1 • X 7n + n n=2 3 1 5n • X 6n − 3 n=1 1 sin2 n + arctan n + e−n • X p 9=5 3 n=1 n + n 1 cos n + 4 + e−n • X p n n=1 4 If a correct inequality cannot be established for the Comparison Test, the following test may be more helpful. 1 1 P P Limit Comparison Test: Suppose an and bn are two series with positive terms. n=1 n=1 a If lim n = L where L > 0 (and is finite), then either BOTH SERIES DIVERGE OR BOTH n!1 bn SERIES CONVERGE. If the limit is 0 or 1, no conclusion can be made. 1 n3 − 2 • X 8n4 + 6n n=5 p 1 n + 3 • X n2 + n − 1 n=1 **The Limit Comparison Test tells us that for rational/algebraic fractions, we can just look at the ratio of the highest powers from the numerator and denominator to determine whether the series converges or diverges.** It is often easier to use than the Comparison Test, especially if you don't want to think about inequalities. 5 1 arctan n • X p n n=1 1 5n • X 6n − 3 n=1 1 1 • X sin n n=1 1 1 • X sin n2 n=1 6 1 P Remainder Estimate for the Integral Test: Suppose we know that the series an converges by using the n=1 Integral Test where f(n) = an. This means we know that the series has a finite sum S. If we use the nth partial sum, sn, to approximate the actual sum S, then the remainder of this estimate is Rn = S − sn, and it satisfies the following inequalities: Z 1 Z 1 f(x) dx ≤ Rn ≤ f(x) dx n+1 n So, if we find that A ≤ Rn ≤ B, then we can say that the actual sum of the series S satisfies sn + A ≤ S ≤ sn + B 1 5 Example: Consider the series X , which is a convergent p-series. n5 n=1 • Find s10 and the upper and lower bounds on the remainder. • Find upper and lower bounds on the actual sum S. • How many terms would be needed to guarantee accuracy to within 0.00001? 7 1 1 Example: Consider the series X . It can be shown this series converges by the Integral Test. n(1 + ln n)2 n=1 • Estimate the error in using s6 to approximate the sum of the series. • How many terms are needed to guarantee the approximation is accurate to within 0.001? 1 sin n + 2 Example: Consider the series X . n3 n=1 • Show the series converges. • Estimate the error in using s5 to approximate the sum of the series. 8 11.5-11.6 Other Convergence Tests In the previous section, the convergence tests we used were for series with positive terms. However, some series do not have all positive terms. An alternating series is a series whose terms alternate positive and negative. Alternating Series Test: 1 P n Suppose you have an alternating series (−1) bn where the bn's are just the positive parts (the absolute n=1 values) of each term. This alternating series converges if: (1) lim bn = 0 n!1 (2) bn+1 ≤ bn for all n. In other words, the absolute values of the terms are decreasing. Notes: • Criteria (1) above is basically just the Test for Divergence. If lim bn is NOT 0, then the limit of the n!1 alternating terms themselves DNE and the series diverges by the Test for Divergence. • As with the Integral Test, criteria (2) just has to be always true eventually. As long as bn+1 ≤ bn is always true after some finite point, we're good to go. Examples: Determine whether the following series converge or diverge. 1 n X (−1) n n=1 1 n−1 2 X (−1) 5n 7n2 + n n=1 9 1 n+1 2 X (−1) n n3 + 4 n=1 1 P n Alternating Series Estimation Theorem: If an alternating series (−1) bn converges, then the error (re- n=1 mainder) in using the nth partial sum sn to approximate the sum of the series satisfies the inequality jRnj ≤ bn+1 Example: 1 (−1)n Consider the convergent alternating series X p . n=1 n + 3 • Estimate the error in using s10 to approximate the sum of the series. • How many terms are needed to approximate the sum to within 0.01? 10 1 (−1)n+1 Example: Approximate the sum of the series X to within 0.00005. 3nn! n=1 P P Absolute Convergence: A series an is said to be absolutely convergent if the series janj is convergent. In other words, if the series of just the absolute values of the terms converges, the series is absolutely convergent. Notes: • It is possible for a series to be convergent, but not absolutely convergent. This is called conditional convergence. • **If a series is absolutely convergent, then it is convergent.** Examples: Determine whether the following series are absolutely convergent, convergent, or divergent. 1 X 1 n3 n=2 1 n X (−1) n n=2 11 1 n X (−1) n n3 + 9 n=3 1 n−1 X (−1) n ln n n=2 1 X cos n 3=2 n=1 n 12 The following is a VERY VERY USEFUL test for absolute convergence, and thus convergence. 1 P an+1 The Ratio Test: For a series an, let lim = L. n=1 n!1 an (i) If L < 1, the series is absolutely convergent, and thus convergent. (ii) If L > 1 or if the limit is infinite, the series is divergent. (iii) If L = 1, the Ratio Test is inconclusive. You cannot say whether the series converges or diverges. Use another test. Notes: • Use the Ratio Test when the series involves factorials and/or exponentials. • The Ratio Test will NOT work if the series involves only algebraic expressions involving powers of n. • The Ratio Test can be used on positive series and/or alternating series. Examples: Determine if the following seres are absolutely convergent, convergent, or divergent. 1 n X (−5) n7n+1 n=1 1 X n! 62n+1 n=1 13 1 n 2 n−1 X (−1) n 8 (3n + 1)! n=1 1 n X (−1) (n + 2) (n + 1)3 n=1 1 X 5 · 9 · 13 ··· (4n + 1) (−1)n+1n4(n + 2)! n=1 14.
Load more

Recommended publications
  • Abel's Identity, 131 Abel's Test, 131–132 Abel's Theorem, 463–464 Absolute Convergence, 113–114 Implication of Condi
    INDEX Abel’s identity, 131 Bolzano-Weierstrass theorem, 66–68 Abel’s test, 131–132 for sequences, 99–100 Abel’s theorem, 463–464 boundary points, 50–52 absolute convergence, 113–114 bounded functions, 142 implication of conditional bounded sets, 42–43 convergence, 114 absolute value, 7 Cantor function, 229–230 reverse triangle inequality, 9 Cantor set, 80–81, 133–134, 383 triangle inequality, 9 Casorati-Weierstrass theorem, 498–499 algebraic properties of Rk, 11–13 Cauchy completeness, 106–107 algebraic properties of continuity, 184 Cauchy condensation test, 117–119 algebraic properties of limits, 91–92 Cauchy integral theorem algebraic properties of series, 110–111 triangle lemma, see triangle lemma algebraic properties of the derivative, Cauchy principal value, 365–366 244 Cauchy sequences, 104–105 alternating series, 115 convergence of, 105–106 alternating series test, 115–116 Cauchy’s inequalities, 437 analytic functions, 481–482 Cauchy’s integral formula, 428–433 complex analytic functions, 483 converse of, 436–437 counterexamples, 482–483 extension to higher derivatives, 437 identity principle, 486 for simple closed contours, 432–433 zeros of, see zeros of complex analytic on circles, 429–430 functions on open connected sets, 430–432 antiderivatives, 361 on star-shaped sets, 429 for f :[a, b] Rp, 370 → Cauchy’s integral theorem, 420–422 of complex functions, 408 consequence, see deformation of on star-shaped sets, 411–413 contours path-independence, 408–409, 415 Cauchy-Riemann equations Archimedean property, 5–6 motivation, 297–300
    [Show full text]
  • Ch. 15 Power Series, Taylor Series
    Ch. 15 Power Series, Taylor Series 서울대학교 조선해양공학과 서유택 2017.12 ※ 본 강의 자료는 이규열, 장범선, 노명일 교수님께서 만드신 자료를 바탕으로 일부 편집한 것입니다. Seoul National 1 Univ. 15.1 Sequences (수열), Series (급수), Convergence Tests (수렴판정) Sequences: Obtained by assigning to each positive integer n a number zn z . Term: zn z1, z 2, or z 1, z 2 , or briefly zn N . Real sequence (실수열): Sequence whose terms are real Convergence . Convergent sequence (수렴수열): Sequence that has a limit c limznn c or simply z c n . For every ε > 0, we can find N such that Convergent complex sequence |zn c | for all n N → all terms zn with n > N lie in the open disk of radius ε and center c. Divergent sequence (발산수열): Sequence that does not converge. Seoul National 2 Univ. 15.1 Sequences, Series, Convergence Tests Convergence . Convergent sequence: Sequence that has a limit c Ex. 1 Convergent and Divergent Sequences iin 11 Sequence i , , , , is convergent with limit 0. n 2 3 4 limznn c or simply z c n Sequence i n i , 1, i, 1, is divergent. n Sequence {zn} with zn = (1 + i ) is divergent. Seoul National 3 Univ. 15.1 Sequences, Series, Convergence Tests Theorem 1 Sequences of the Real and the Imaginary Parts . A sequence z1, z2, z3, … of complex numbers zn = xn + iyn converges to c = a + ib . if and only if the sequence of the real parts x1, x2, … converges to a . and the sequence of the imaginary parts y1, y2, … converges to b. Ex.
    [Show full text]
  • Topic 7 Notes 7 Taylor and Laurent Series
    Topic 7 Notes Jeremy Orloff 7 Taylor and Laurent series 7.1 Introduction We originally defined an analytic function as one where the derivative, defined as a limit of ratios, existed. We went on to prove Cauchy's theorem and Cauchy's integral formula. These revealed some deep properties of analytic functions, e.g. the existence of derivatives of all orders. Our goal in this topic is to express analytic functions as infinite power series. This will lead us to Taylor series. When a complex function has an isolated singularity at a point we will replace Taylor series by Laurent series. Not surprisingly we will derive these series from Cauchy's integral formula. Although we come to power series representations after exploring other properties of analytic functions, they will be one of our main tools in understanding and computing with analytic functions. 7.2 Geometric series Having a detailed understanding of geometric series will enable us to use Cauchy's integral formula to understand power series representations of analytic functions. We start with the definition: Definition. A finite geometric series has one of the following (all equivalent) forms. 2 3 n Sn = a(1 + r + r + r + ::: + r ) = a + ar + ar2 + ar3 + ::: + arn n X = arj j=0 n X = a rj j=0 The number r is called the ratio of the geometric series because it is the ratio of consecutive terms of the series. Theorem. The sum of a finite geometric series is given by a(1 − rn+1) S = a(1 + r + r2 + r3 + ::: + rn) = : (1) n 1 − r Proof.
    [Show full text]
  • 3.3 Convergence Tests for Infinite Series
    3.3 Convergence Tests for Infinite Series 3.3.1 The integral test We may plot the sequence an in the Cartesian plane, with independent variable n and dependent variable a: n X The sum an can then be represented geometrically as the area of a collection of rectangles with n=1 height an and width 1. This geometric viewpoint suggests that we compare this sum to an integral. If an can be represented as a continuous function of n, for real numbers n, not just integers, and if the m X sequence an is decreasing, then an looks a bit like area under the curve a = a(n). n=1 In particular, m m+2 X Z m+1 X an > an dn > an n=1 n=1 n=2 For example, let us examine the first 10 terms of the harmonic series 10 X 1 1 1 1 1 1 1 1 1 1 = 1 + + + + + + + + + : n 2 3 4 5 6 7 8 9 10 1 1 1 If we draw the curve y = x (or a = n ) we see that 10 11 10 X 1 Z 11 dx X 1 X 1 1 > > = − 1 + : n x n n 11 1 1 2 1 (See Figure 1, copied from Wikipedia) Z 11 dx Now = ln(11) − ln(1) = ln(11) so 1 x 10 X 1 1 1 1 1 1 1 1 1 1 = 1 + + + + + + + + + > ln(11) n 2 3 4 5 6 7 8 9 10 1 and 1 1 1 1 1 1 1 1 1 1 1 + + + + + + + + + < ln(11) + (1 − ): 2 3 4 5 6 7 8 9 10 11 Z dx So we may bound our series, above and below, with some version of the integral : x If we allow the sum to turn into an infinite series, we turn the integral into an improper integral.
    [Show full text]
  • The Binomial Series 16.3   
    The Binomial Series 16.3 Introduction In this Section we examine an important example of an infinite series, the binomial series: p(p − 1) p(p − 1)(p − 2) 1 + px + x2 + x3 + ··· 2! 3! We show that this series is only convergent if |x| < 1 and that in this case the series sums to the value (1 + x)p. As a special case of the binomial series we consider the situation when p is a positive integer n. In this case the infinite series reduces to a finite series and we obtain, by replacing x with b , the binomial theorem: a n(n − 1) (b + a)n = bn + nbn−1a + bn−2a2 + ··· + an. 2! Finally, we use the binomial series to obtain various polynomial expressions for (1 + x)p when x is ‘small’. ' • understand the factorial notation $ Prerequisites • have knowledge of the ratio test for convergence of infinite series. Before starting this Section you should ... • understand the use of inequalities '& • recognise and use the binomial series $% Learning Outcomes • state and use the binomial theorem On completion you should be able to ... • use the binomial series to obtain numerical approximations & % 26 HELM (2008): Workbook 16: Sequences and Series ® 1. The binomial series A very important infinite series which occurs often in applications and in algebra has the form: p(p − 1) p(p − 1)(p − 2) 1 + px + x2 + x3 + ··· 2! 3! in which p is a given number and x is a variable. By using the ratio test it can be shown that this series converges, irrespective of the value of p, as long as |x| < 1.
    [Show full text]
  • 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 Terminology
    AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential
    [Show full text]
  • Learning Goals: Power Series • Master the Art of Using the Ratio Test
    Learning Goals: Power Series • Master the art of using the ratio test to find the radius of convergence of a power series. • Learn how to find the series associated with the end points of the interval of convergence. 1 X xn • Become familiar with the series : n! n=0 xn • Become familiar with the limit lim : n!1 n! • Know the Power series representation of ex • Learn how to manipulate the power series representation of ex using tools previously mastered such as substitution, integration and differentiation. • Learn how to use power series to calculate limits. 1 Power Series, Stewart, Section 11.8 In this section, we will use the ratio test to determine the Radius of Convergence of a given power series. We will also determine the interval of convergence of the given power series if this is possible using the tests for convergence that we have already developed, namely, recognition as a geometric series, harmonic series or alternating harmonic series, telescoping series, the divergence test along with the fact that a series is absolutely convergent converges. The ratio and root tests will be inconclusive at the end points of the interval of convergence in the examples below since we use them to determine the radius of convergence. We may not be able to decide on convergence or divergence for some of the end points with our current tools. We will expand our methods for testing individual series for convergence in subsequent sections and tie up any loose ends as we proceed. We recall the definition of the Radius of Convergence and Interval of Convergence of a power series below.
    [Show full text]
  • Sequences, Series and Taylor Approximation (Ma2712b, MA2730)
    Sequences, Series and Taylor Approximation (MA2712b, MA2730) Level 2 Teaching Team Current curator: Simon Shaw November 20, 2015 Contents 0 Introduction, Overview 6 1 Taylor Polynomials 10 1.1 Lecture 1: Taylor Polynomials, Definition . .. 10 1.1.1 Reminder from Level 1 about Differentiable Functions . .. 11 1.1.2 Definition of Taylor Polynomials . 11 1.2 Lectures 2 and 3: Taylor Polynomials, Examples . ... 13 x 1.2.1 Example: Compute and plot Tnf for f(x) = e ............ 13 1.2.2 Example: Find the Maclaurin polynomials of f(x) = sin x ...... 14 2 1.2.3 Find the Maclaurin polynomial T11f for f(x) = sin(x ) ....... 15 1.2.4 QuestionsforChapter6: ErrorEstimates . 15 1.3 Lecture 4 and 5: Calculus of Taylor Polynomials . .. 17 1.3.1 GeneralResults............................... 17 1.4 Lecture 6: Various Applications of Taylor Polynomials . ... 22 1.4.1 RelativeExtrema .............................. 22 1.4.2 Limits .................................... 24 1.4.3 How to Calculate Complicated Taylor Polynomials? . 26 1.5 ExerciseSheet1................................... 29 1.5.1 ExerciseSheet1a .............................. 29 1.5.2 FeedbackforSheet1a ........................... 33 2 Real Sequences 40 2.1 Lecture 7: Definitions, Limit of a Sequence . ... 40 2.1.1 DefinitionofaSequence .......................... 40 2.1.2 LimitofaSequence............................. 41 2.1.3 Graphic Representations of Sequences . .. 43 2.2 Lecture 8: Algebra of Limits, Special Sequences . ..... 44 2.2.1 InfiniteLimits................................ 44 1 2.2.2 AlgebraofLimits.............................. 44 2.2.3 Some Standard Convergent Sequences . .. 46 2.3 Lecture 9: Bounded and Monotone Sequences . ..... 48 2.3.1 BoundedSequences............................. 48 2.3.2 Convergent Sequences and Closed Bounded Intervals . .... 48 2.4 Lecture10:MonotoneSequences .
    [Show full text]
  • Calculus Online Textbook Chapter 10
    Contents CHAPTER 9 Polar Coordinates and Complex Numbers 9.1 Polar Coordinates 348 9.2 Polar Equations and Graphs 351 9.3 Slope, Length, and Area for Polar Curves 356 9.4 Complex Numbers 360 CHAPTER 10 Infinite Series 10.1 The Geometric Series 10.2 Convergence Tests: Positive Series 10.3 Convergence Tests: All Series 10.4 The Taylor Series for ex, sin x, and cos x 10.5 Power Series CHAPTER 11 Vectors and Matrices 11.1 Vectors and Dot Products 11.2 Planes and Projections 11.3 Cross Products and Determinants 11.4 Matrices and Linear Equations 11.5 Linear Algebra in Three Dimensions CHAPTER 12 Motion along a Curve 12.1 The Position Vector 446 12.2 Plane Motion: Projectiles and Cycloids 453 12.3 Tangent Vector and Normal Vector 459 12.4 Polar Coordinates and Planetary Motion 464 CHAPTER 13 Partial Derivatives 13.1 Surfaces and Level Curves 472 13.2 Partial Derivatives 475 13.3 Tangent Planes and Linear Approximations 480 13.4 Directional Derivatives and Gradients 490 13.5 The Chain Rule 497 13.6 Maxima, Minima, and Saddle Points 504 13.7 Constraints and Lagrange Multipliers 514 CHAPTER Infinite Series Infinite series can be a pleasure (sometimes). They throw a beautiful light on sin x and cos x. They give famous numbers like n and e. Usually they produce totally unknown functions-which might be good. But on the painful side is the fact that an infinite series has infinitely many terms. It is not easy to know the sum of those terms.
    [Show full text]
  • The Ratio Test, Integral Test, and Absolute Convergence
    MATH 1D, WEEK 3 { THE RATIO TEST, INTEGRAL TEST, AND ABSOLUTE CONVERGENCE INSTRUCTOR: PADRAIC BARTLETT Abstract. These are the lecture notes from week 3 of Ma1d, the Caltech mathematics course on sequences and series. 1. Homework 1 data • HW average: 91%. • Comments: none in particular { people seemed to be pretty capable with this material. 2. The Ratio Test So: thus far, we've developed a few tools for determining the convergence or divergence of series. Specifically, the main tools we developed last week were the two comparison tests, which gave us a number of tools for determining whether a series converged by comparing it to other, known series. However, sometimes this P 1 P n isn't enough; simply comparing things to n and r usually can only get us so far. Hence, the creation of the following test: Theorem 2.1. If fang is a sequence of positive numbers such that a lim n+1 = r; n!1 an for some real number r, then P1 • n=1 an converges if r < 1, while P1 • n=1 an diverges if r > 1. If r = 1, this test is inconclusive and tells us nothing. Proof. The basic idea motivating this theorem is the following: if the ratios an+1 an eventually approach some value r, then this series eventually \looks like" the geo- metric series P rn; consequently, it should converge whenever r < 1 and diverge whenever r > 1. an+1 To make the above concrete: suppose first that r > 1. Then, because limn!1 = an r;, we know that there is some N 2 N such that 8n ≥ N, a n+1 > 1 an )an+1 > an: But this means that the an's are eventually an increasing sequence starting at some value aN > 0; thus, we have that limn!1 an 6= 0 and thus that the sum P1 n=1 an cannot converge.
    [Show full text]
  • 1 Convergence Tests
    Lecture 24Section 11.4 Absolute and Conditional Convergence; Alternating Series Jiwen He 1 Convergence Tests Basic Series that Converge or Diverge Basic Series that Converge X Geometric series: xk, if |x| < 1 X 1 p-series: , if p > 1 kp Basic Series that Diverge X Any series ak for which lim ak 6= 0 k→∞ X 1 p-series: , if p ≤ 1 kp Convergence Tests (1) Basic Test for Convergence P Keep in Mind that, if ak 9 0, then the series ak diverges; therefore there is no reason to apply any special convergence test. P k P k k Examples 1. x with |x| ≥ 1 (e.g, (−1) ) diverge since x 9 0. [1ex] k X k X 1 diverges since k → 1 6= 0. [1ex] 1 − diverges since k + 1 k+1 k 1 k −1 ak = 1 − k → e 6= 0. Convergence Tests (2) Comparison Tests Rational terms are most easily handled by basic comparison or limit comparison with p-series P 1/kp Basic Comparison Test 1 X 1 X 1 X k3 converges by comparison with converges 2k3 + 1 k3 k5 + 4k4 + 7 X 1 X 1 X 2 by comparison with converges by comparison with k2 k3 − k2 k3 X 1 X 1 X 1 diverges by comparison with diverges by 3k + 1 3(k + 1) ln(k + 6) X 1 comparison with k + 6 Limit Comparison Test X 1 X 1 X 3k2 + 2k + 1 converges by comparison with . diverges k3 − 1 √ k3 k3 + 1 X 3 X 5 k + 100 by comparison with √ √ converges by comparison with k 2k2 k − 9 k X 5 2k2 Convergence Tests (3) Root Test and Ratio Test The root test is used only if powers are involved.
    [Show full text]