DOCSLIB.ORG

Strategy for Testing Series

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Strategy for Testing Series Strategy for Testing Series We now have several ways of testing a series for convergence or divergence; the problem is to decide which test to use on which series. In this respect testing series is similar to inte- grating functions. Again there are no hard and fast rules about which test to apply to a given series, but you may find the following advice of some use. It is not wise to apply a list of the tests in a specific order until one finally works. That would be a waste of time and effort. Instead, as with integration, the main strategy is to classify the series according to its form. 1. If the series is of the form ͸ 1͞n p, it is a p-series, which we know to be conver- gent if p Ͼ 1 and divergent if p ഛ 1. Ϫ 2. If the series has the form ͸ ar n 1 or ͸ ar n, it is a geometric series, which con- verges if Խ r Խ Ͻ 1 and diverges if Խ r Խ ജ 1. Some preliminary algebraic manipula- tion may be required to bring the series into this form. 3. If the series has a form that is similar to a p-series or a geometric series, then one of the comparison tests should be considered. In particular, if an is a rational function or algebraic function of n (involving roots of polynomials), then the series should be compared with a p-series. (The value of p should be chosen as in Section 8.3 by keeping only the highest powers of n in the numerator and denominator.) The comparison tests apply only to series with positive terms, but if ͸ a has some negative terms, then we can apply the Comparison Test to ͸ n Խ an Խ and test for absolute convergence. 4. If you can see at a glance that lim n l ϱ an 0, then the Test for Divergence should be used. ͸ nϪ1 ͸ n 5. If the series is of the form ͑Ϫ1͒ bn or ͑Ϫ1͒ bn, then the Alternating Series Test is an obvious possibility. 6. Series that involve factorials or other products (including a constant raised to the nth power) are often conveniently tested using the Ratio Test. Bear in mind that Խ anϩ1͞an Խl 1 as n l ϱ for all p-series and therefore all rational or algebraic functions of n. Thus, the Ratio Test should not be used for such series. ෇ ͑ ͒ xϱ ͑ ͒ 7. If an f n , where 1 f x dx is easily evaluated, then the Integral Test is effec- tive (assuming the hypotheses of this test are satisfied). In the following examples we don’t work out all the details but simply indicate which tests should be used. ϱ n Ϫ 1 EXAMPLE 1 ͚ n෇1 2n ϩ 1 l 1 l ϱ Since an 2 0 as n , we should use the Test for Divergence. ϱ sn 3 ϩ 1 EXAMPLE 2 ͚ 3 2 n෇1 3n ϩ 4n ϩ 2 Since a is an algebraic function of n, we compare the given series with a p-series. n ͸ The comparison series for the Limit Comparison Test is bn, where sn 3 n 3͞2 1 b ෇ ෇ ෇ n 3n 3 3n 3 3n 3͞2 1 2 ■ STRATEGY FOR TESTING SERIES ϱ Ϫ 2 EXAMPLE 3 ͚ ne n n෇1 xϱ Ϫx2 Since the integral 1 xe dx is easily evaluated, we use the Integral Test. The Ratio Test also works. ϱ n 3 ͑Ϫ ͒n EXAMPLE 4 ͚ 1 4 n෇1 n ϩ 1 Since the series is alternating, we use the Alternating Series Test. ϱ 2k EXAMPLE 5 ͚ k෇1 k! Since the series involves k!, we use the Ratio Test. ϱ 1 EXAMPLE 6 ͚ n n෇1 2 ϩ 3 Since the series is closely related to the geometric series ͸ 1͞3n, we use the Comparison Test. Exercises ϱ ϱ ͑Ϫ ͒nϪ1 ͞ 1 A Click here for answers. S Click here for solutions. 17. ͚ ͑Ϫ1͒n21 n 18. ͚ n෇1 n෇2 sn Ϫ 1 ϱ ϱ 1–34 Test the series for convergence or divergence. ln n k ϩ 5 19. ͚ ͑Ϫ1͒n 20. ͚ ϱ ϱ s k n 2 Ϫ 1 n Ϫ 1 n෇1 n k෇1 5 1. ͚ 2 2. ͚ 2 ϱ 2n ϱ 2 ෇ n ϩ n ෇ n ϩ n ͑Ϫ2͒ sn Ϫ 1 n 1 n 1 21. ͚ 22. ͚ n 3 ϩ 2 ϩ ϱ ϱ Ϫ n෇1 n n෇1 n 2n 5 1 nϪ1 n 1 3. ͚ 4. ͚ ͑Ϫ1͒ ϱ ϱ 2 ϩ 2 ϩ cos͑n͞2͒ n෇1 n n n෇1 n n ͑ ͞ ͒ 23. ͚ tan 1 n 24. ͚ 2 ෇ ෇ n ϩ 4n ϱ ͑Ϫ3͒nϩ1 ϱ 1 n 1 n 1 ϱ ϱ 2 5. ͚ 3n 6. ͚ 2 n! n ϩ 1 n෇1 2 n෇1 n ϩ n cos n 25. ͚ n2 26. ͚ n n෇1 e n෇1 5 ϱ 1 ϱ 2 k k! 7. ͚ 8. ͚ ϱ k ln k ϱ e 1͞n n෇2 nsln n k෇1 ͑k ϩ 2͒! 27. ͚ 3 28. ͚ 2 k෇1 ͑k ϩ 1͒ n෇1 n ϱ ϱ 2 Ϫk 2 Ϫn3 ϱ Ϫ1 ϱ s 9. ͚ k e 10. ͚ n e tan n j j k෇1 n෇1 29. ͚ 30. ͚ ͑Ϫ1͒ n෇1 nsn j෇1 j ϩ 5 ϱ nϩ1 ϱ ͑Ϫ1͒ n ϱ k ϱ ͑Ϫ ͒n 5 1 11. ͚ 12. ͚ 1 2 n෇2 n ln n n෇1 n ϩ 25 31. ͚ k k 32. ͚ ln n k෇1 3 ϩ 4 n෇2 ͑ln n͒ ϱ n 2 ϱ 3 n ϱ sin͑1͞n͒ ϱ 13. ͚ 14. ͚ sin n 33. ͚ 34. ͚ (sn 2 Ϫ 1) n෇1 n! n෇1 n෇1 sn n෇1 ϱ n! 15. ͚ n෇0 2 ؒ 5 ؒ 8 ؒ иии ؒ ͑3n ϩ 2͒ ϱ n 2 ϩ 1 16. ͚ 3 n෇1 n ϩ 1 STRATEGY FOR TESTING SERIES ■ 3 Answers S Click here for solutions. 1. D 3. C 5. C 7. D 9. C 11. C 13. C 15. C 17. D 19. C 21. C 23. D 25. C 27. C 29. C 31. D 33. C 4 ■ STRATEGY FOR TESTING SERIES Solutions: Strategy for Testing Series n2 1 1 1/n2 ∞ n2 1 1. lim an =lim − =lim − =1=0,sotheseries − diverges by the Test for n n n2 +1 n 1+1/n 6 n2 +1 →∞ →∞ →∞ n=1 X Divergence. 1 1 ∞ 1 ∞ 1 3. 2 < 2 for all n 1,so 2 converges by the Comparison Test with 2 ,ap-series that n + n n ≥ n=1 n + n n=1 n converges because p =2> 1. P P n+2 3n 3n an+1 ( 3) 2 3 2 3 3 5. lim =lim −3(n+1) n+1 =lim −3n· 3 =lim 3 = < 1,sotheseries n an n 2 · ( 3) n 2 2 n 2 8 →∞ ¯ ¯ →∞ ¯ − ¯ →∞ ¯ · ¯ →∞ ¯ n+1¯ ¯ ¯ ¯ ¯ ∞ (¯ 3) ¯ ¯ ¯ ¯ ¯ −¯ ¯ is absolutely¯ convergent by the¯ Ratio Test.¯ ¯ 23n n=1 X 1 7. Let f(x)= .Thenf is positive, continuous, and decreasing on [2, ), so we can apply the Integral Test. x √ln x ∞ 1 u =lnx, 1/2 1/2 Since dx = u− du =2u + C =2√ln x + C,wefind x√ln x du = dx/x Z " # Z t ∞ dx dx t =lim =lim 2√ln x =lim 2 √ln t 2 √ln 2 = .Sincetheintegral 2 x √ln x t 2 x √ln x t 2 t − ∞ Z →∞ Z →∞ h i →∞ ³ ´ ∞ 1 diverges, the given series diverges. √ n=2 n ln n X 2 ∞ 2 k ∞ k 9. k e− = . Using the Ratio Test, we get ek k=1 k=1 X X 2 k 2 ak+1 (k +1) e k +1 1 2 1 1 lim =lim k+1 2 =lim =1 = < 1, so the series converges. k ak k e · k k k · e · e e →∞ ¯ ¯ →∞ ¯ ¯ →∞ "µ ¶ # ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ n+1 ¯ 1 ¯ ¯ ¯ ∞ ( 1) 11. bn = > 0 for n 2, bn is decreasing, and lim bn =0,sothegivenseries − converges by n ln n n n ln n ≥ { } →∞ n=2 the Alternating Series Test. P n+1 2 2 an+1 3 (n +1) n! 3(n +1) n +1 13. lim =lim n 2 =lim 2 =3 lim 2 =0< 1,sotheseries n an n (n +1)! · 3 n n (n +1)n n n →∞ ¯ ¯ →∞ ¯ ¯ →∞ · ¸ →∞ ¯n 2 ¯ ¯ ¯ ∞ 3¯ n ¯ ¯ ¯ ¯ converges¯ by¯ the Ratio Test. ¯ n! n=1 X an+1 (n +1)! 2 5 8 (3n +2) 15. lim =lim · · ····· n an n 2 5 8 (3n +2)[3(n +1)+2] · n! →∞ ¯ ¯ →∞ ¯ · · ····· ¯ ¯ ¯ ¯n +1 1 ¯ ¯ ¯=lim¯ = < 1 ¯ ¯ ¯ n ¯3n +5 3 ¯ →∞ ∞ n! so the series converges by the Ratio Test. 2 5 8 (3n +2) n=0 X · · ····· ∞ 17. lim 21/n =20 =1,so lim ( 1)n 21/n does not exist and the series ( 1)n21/n diverges by the n n →∞ →∞ − n=1 − X Test for Divergence. STRATEGY FOR TESTING SERIES ■ 5 ln x 2 ln x 2 ln n 2 19. Let f(x)= .Thenf 0(x)= − < 0 when ln x>2 or x>e ,so is decreasing for n>e . √x 2x3/2 √n ln n 1/n 2 ∞ ln n By l’Hospital’s Rule, lim = lim = lim =0,sotheseries ( 1)n converges by n √n n 1/ (2√n) n √n √n →∞ →∞ →∞ n=1 − X the Alternating Series Test.
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]
  • 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.
    [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]
  • Curl, Divergence and Laplacian
    Curl, Divergence and Laplacian What to know: 1. The definition of curl and it two properties, that is, theorem 1, and be able to predict qualitatively how the curl of a vector field behaves from a picture. 2. The definition of divergence and it two properties, that is, if div F~ 6= 0 then F~ can't be written as the curl of another field, and be able to tell a vector field of clearly nonzero,positive or negative divergence from the picture. 3. Know the definition of the Laplace operator 4. Know what kind of objects those operator take as input and what they give as output. The curl operator Let's look at two plots of vector fields: Figure 1: The vector field Figure 2: The vector field h−y; x; 0i: h1; 1; 0i We can observe that the second one looks like it is rotating around the z axis. We'd like to be able to predict this kind of behavior without having to look at a picture. We also promised to find a criterion that checks whether a vector field is conservative in R3. Both of those goals are accomplished using a tool called the curl operator, even though neither of those two properties is exactly obvious from the definition we'll give. Definition 1. Let F~ = hP; Q; Ri be a vector field in R3, where P , Q and R are continuously differentiable. We define the curl operator: @R @Q @P @R @Q @P curl F~ = − ~i + − ~j + − ~k: (1) @y @z @z @x @x @y Remarks: 1.
    [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 Divergence of a Vector Field.Doc 1/8
    9/16/2005 The Divergence of a Vector Field.doc 1/8 The Divergence of a Vector Field The mathematical definition of divergence is: w∫∫ A(r )⋅ds ∇⋅A()rlim = S ∆→v 0 ∆v where the surface S is a closed surface that completely surrounds a very small volume ∆v at point r , and where ds points outward from the closed surface. From the definition of surface integral, we see that divergence basically indicates the amount of vector field A ()r that is converging to, or diverging from, a given point. For example, consider these vector fields in the region of a specific point: ∆ v ∆v ∇⋅A ()r0 < ∇ ⋅>A (r0) Jim Stiles The Univ. of Kansas Dept. of EECS 9/16/2005 The Divergence of a Vector Field.doc 2/8 The field on the left is converging to a point, and therefore the divergence of the vector field at that point is negative. Conversely, the vector field on the right is diverging from a point. As a result, the divergence of the vector field at that point is greater than zero. Consider some other vector fields in the region of a specific point: ∇⋅A ()r0 = ∇ ⋅=A (r0) For each of these vector fields, the surface integral is zero. Over some portions of the surface, the normal component is positive, whereas on other portions, the normal component is negative. However, integration over the entire surface is equal to zero—the divergence of the vector field at this point is zero. * Generally, the divergence of a vector field results in a scalar field (divergence) that is positive in some regions in space, negative other regions, and zero elsewhere.
    [Show full text]
  • Generalized Stokes' Theorem
    Chapter 4 Generalized Stokes’ Theorem “It is very difficult for us, placed as we have been from earliest childhood in a condition of training, to say what would have been our feelings had such training never taken place.” Sir George Stokes, 1st Baronet 4.1. Manifolds with Boundary We have seen in the Chapter 3 that Green’s, Stokes’ and Divergence Theorem in Multivariable Calculus can be unified together using the language of differential forms. In this chapter, we will generalize Stokes’ Theorem to higher dimensional and abstract manifolds. These classic theorems and their generalizations concern about an integral over a manifold with an integral over its boundary. In this section, we will first rigorously define the notion of a boundary for abstract manifolds. Heuristically, an interior point of a manifold locally looks like a ball in Euclidean space, whereas a boundary point locally looks like an upper-half space. n 4.1.1. Smooth Functions on Upper-Half Spaces. From now on, we denote R+ := n n f(u1, ... , un) 2 R : un ≥ 0g which is the upper-half space of R . Under the subspace n n n topology, we say a subset V ⊂ R+ is open in R+ if there exists a set Ve ⊂ R open in n n n R such that V = Ve \ R+. It is intuitively clear that if V ⊂ R+ is disjoint from the n n n subspace fun = 0g of R , then V is open in R+ if and only if V is open in R . n n Now consider a set V ⊂ R+ which is open in R+ and that V \ fun = 0g 6= Æ.
    [Show full text]