Infinite Series In Calculus

Infinite Series In Calculus In the study of calculus, the topic of infinite series generally occurs near the end of the second semester in a typical two-semester sequence in single variable calculus. This seems to be one of the most difficult topics for students to understand and for teachers to explain clearly. It should not be surprising that these ideas are as difficult to grasp as the use of ε−δ proofs for limits, since these concepts—limits and infinite series—are related, and both took hundreds of years to formulate. Thus, even though now we have a better perspective from which to start, it is still much to ask of our students to gain a full understanding of infinite series in the two to four weeks given to their study in a beginning calculus course. Some History of Infinite Series Concepts surrounding infinite series were present in ancient Greek mathematics as Zeno, Archimedes, and other mathematicians worked with finite sums. Zeno posed his paradox in about 450 BCE, and Archimedes found the area of a parabolic segment in approximately 250 BCE by determining the sum of the infinite geometric series with constant ratio 1 4 (Stillwell 1989, 170). One cannot credit Archimedes (or the Greeks) with discovering infinite series, since Archimedes worked only with finite sums and determined that certain finite sums underapproximated the area and others overapproximated the area, leaving the common limit (not known to him as a limit) as the area of the parabolic segment. However, this does point out one of the two motivations for the development of infinite series: (1) to approximate unknown areas, and (2) to approximate the value of π. A nongeometric series appeared in Liber calculationum by Richard Suiseth, known as “The Calculator” (Stillwell 1989, 118), in approximately 1350. Suiseth Special Focus: Calculus 1 2 3 k + 2 + 3 + L + k + L = 2 . At about the same time, Nicole Oresme, 2 2 2 2 a bishop in Normandy (Cajori 1919, 127), also found this infinite sum along with indicated that similar ones, and proved the divergence of the harmonic series using 1+ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 + + + + + + +L > 1+ + + + + + + +L 2 3 4 5 6 7 8 2 4 4 8 8 8 8 However, in the same century, Madhava of Sangamagramma (c. 1340–1425) and fellow scholars of the Kerala school in southern India were making even more important discoveries about infinite series. Madhava, an astronomer and π 1 1 1 mathematician, used the now easily obtained series = 1– + – + L to estimate 4 3 5 7 π, and, in addition, transformed this series into the more rapidly converging series 1 1 1 − 12  1 − + + L for π (Joseph, 290). More generally, he discovered what  3 ⋅ 3 5 ⋅ 3 2 7 ⋅ 33  we now call the Taylor series for arctangent, sine, and cosine (Joseph 2000, 289–293; Katz 2004, 152–156). One theory is that these ideas may have been carried to Europe by Jesuit missionaries to India (Katz 2004, 156). Moving to Europe, Portuguese mathematician Alvarus Thomas considered geometric series in 1509 (Cajori 1919, 172). Pietro Mengoli of Bologna treated particular ∞ 1 infinite series in Novæ quadraturae arithmeticae in 1650, finding ∑ along n = 1 n ( n + 1) with proving the divergence of the harmonic series. In 1668, the theory of power series began with the publication of the series for ln (1 + x ) by Nicolaus Mercator, who did 1 this by “integrating” (Stillwell 1989, 120). 1+ x Newton’s general binomial theorem in 1665 aided the finding of series for many functions. Newton, Gregory, and Leibniz all used interpolation ideas to lead to their important results and went from finite approximations to infinite expansions. There is strong evidence that James Gregory, who was the first to publish a proof of the Fundamental Theorem of Calculus in 1668, used Taylor series in 1671, 44 years prior to Brook Taylor’s results in 1715. In 1734, Leonhard Euler gave new life to infinite series by finding that 1 π 2, ∑ n2 = 6 n= 1 ∞ after attempts by Mengoli and the Bernoulli brothers had failed. In 1748, Euler used 2 Infinite Series In Calculus some ideas from his work in 1734 to generate what we now call the Riemann zeta ∞ 1 z . This function, named after Bernhard Riemann because he n=1 n function ζ ( z ) = ∑ was the first to use complex numbers in the domain of this function, has become important because of its relationship to the distribution of prime numbers and holds the distinction today of pertaining to one of the most famous unsolved problems in mathematics, the Riemann Hypothesis. During this time, issues of convergence of series were barely considered, which often led to confusing and conflicting statements concerning infinite series. The first important and rigorous treatment of infinite series was given by Karl Friedrich Gauss in his study of hypergeometric series in 1812 (Cajori 1919, 373). In 1816, Bernard Bolzano exhibited clear notions of convergence. Augustin-Louis Cauchy shared these ideas with the public in 1821 in his textbook Cours d’analyse de l’École Polytechnique. The ratio and root tests and the idea of absolute convergence were included in this text. Uniform convergence was studied in the middle of the nineteenth century, and divergent series were studied in the late nineteenth century. Today, infinite series are taught in beginning and advanced calculus courses. They are heavily used in the study of differential equations. They are still used to approximate π as illustrated by the BBP (Bailey, Borwein, and Plouffe) formulas. One of these, discovered by Plouffe in 1995, gives the base 16-digit extraction algorithm for π 4 2 1 1  1 n  − − − using π = ∑  . 8 n + 4 8 n + 5 8 n + 6   16  n= 0 8 n + 1 ∞ This historical background would not be complete without mentioning Fourier series, which attempt to give values for a function using an infinite sum of trigonometric functions. These are named after Joseph Fourier, a scientist and mathematician who used them in La Théorie Analytique de la Chaleur in 1822 to study the conduction of heat. You can see that the development of the concepts surrounding infinite series took a long time and involved many different mathematicians with many different ideas. It’s not surprising that our students don’t fully comprehend their nuances in just one semester of study. The Heart of Infinite Series At the heart of infinite series are three concepts: (1) the definition of convergence of an infinite series, 3 Special Focus: Calculus (2) positive term series, and (3) absolute convergence of series. While absolute convergence does not appear specifically in the syllabus, power series cannot be fully considered without this idea, nor can one take fulladvantage of the ratio and root tests. Given students’ difficulty with understanding the concept of infinite series, I believe we frequently “rush” to get to the tests for convergence and never really require students to fully understand the notion of convergence. More time should be spent with students computing partial sums and attempting to find the limits of these sequences of partial sums. For positive term series, convergence of the sequence of partial sums is simple. Since for a positive term series the sequence of partial sums is nondecreasing, convergence of the sequence of partial sums occurs if and only if that sequence is bounded above. We probably should spend more time finding upper bounds for the sequence of partial sums of a positive term series or showing that there is no such upper bound. Nearly all of the convergence tests are founded on this one idea. The comparison test, limit comparison test, and integral test all lead directly to upper bounds for the sequence of partial sums or show that there is no such upper bound. Even the ratio and roots tests essentially are a limit comparison test with a geometric series, and show convergence if the comparison is with a geometric series whose common ratio has an absolute value of less than 1. Thus, the ratio and root tests are just formalized versions of a limit comparison test with a geometric series. Relating these tests back to upper bounds for the sequence of partial sums might help our students see the one common thread for all these tests. For series that have both positive and negative terms, the idea of absolute convergence becomes helpful. If a series converges absolutely, then it must converge. 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 of a particular type of infinite series. I believe we sometimes overemphasize the importance of this test because we want to make clear the distinction between absolute convergence and convergence. That is, we want to give examples of series that converge but do not converge absolutely. It is relatively difficult to do that without giving examples of series satisfying the hypotheses of the alternating series test. This is because it is difficult to show that a series not satisfying the hypotheses is convergent when it is not absolutely convergent. In fact, showing convergence of an arbitrary series can be quite difficult. No matter what rules we might develop to 4 Infinite Series In Calculus determine the convergence of a series, “a series can be invented for which the rule fails to give a decisive result.” (Bromwich 1965, 46). Thus, it is not surprising that infinite series is a difficult topic for our students. Once students understand the concept of convergence and divergence of infinite series, there are two basic questions one can ask about a specific series: (1) Does that series converge? (2) If it converges, to what does it converge; and if it diverges, why? We have already talked about the first question above. The second question really leads to the study of power series. Power series define functions on their intervals of convergence, and the challenge is to identify these functions. The other role of power series is to allow us to express a given function as a power series and then use that expression as a means to approximate the function with a polynomial function, which uses only simple arithmetic (addition and multiplication) to approximate functional values. Polynomial Approximations and Series Concept of series A series is defined as a sequence of partial sums, and convergence is defined in terms of the limit of the sequence of partial sums. Technology can be used to explore convergence and divergence. Series of constants • • • • • • • Motivating examples, including decimal expansion Geometric series with applications The harmonic series Alternating series with error bound Terms of series as areas of rectangles and their relationship to improper integrals, including the integral test and its use in testing the convergence of p-series The ratio test for convergence and divergence Comparing series to test for convergence or divergence 5 Special Focus: Calculus Taylor series • • • • • • • Taylor polynomial approximation with graphical demonstration of convergence (for example, viewing graphs of various Taylor polynomials of the sine function approximating the sine curve) Maclaurin series and the general Taylor series centered at x = a 1 Maclaurin series for the functions ex, sin(x), cos(x), and 1− x Formal manipulation of Taylor series and shortcuts to computing Taylor series, including substitution, differentiation, antidifferentiation, and the formation of new series from known series Functions defined by power series Radius and interval of convergence of power series Lagrange error bound for Taylor polynomials These indicate both what students should know about infinite series and what they should be able to do with infinite series. It’s possible to see these items being tested by examining those free-response questions that have appeared on the Calculus BC Exam over the years. The following table identifies questions pertaining to the particular ideas mentioned in the syllabus using the Calculus BC Examsfrom 1969 to the present. Questions can occur under multiple sections below. 6 Topic Exam Questions Geometric Series 1981-3a, 1981-3b, 1981-3c, 2001-6d, 2002-6c Harmonic Series 1972-4b, 1975-4b, 1982-5b, 2002-6a, 2005-6c Alternating Series Test 1970-6b, 1972-4b, 1975-4b, 1982-5b, 2002-6a Alternating Series Error 1970-6c, 1971-4d, 1979-4c, 1982-5c, 1984-4c, 1990-5c, 1994-5c, 2000-3c, 2003-6b, 2006-6d, 2007-6d Integral Comparisons 1969-7a Integral Test 1969-7bc,1973-6c, 1992-6b Ratio Test 1975-4a nth Term Test (Divergence) 1973-6a Comparison Test 1972-4a, 1973-6b, 1977-5a, 1980-3a, 1980-3c, 1992-6a, 1992-6c Taylor Polynomials 1995-4c, 1997-2a, 1998-3a, 1999-4a, 2000-3a, 2004-6a, 2005-6a Taylor Polynomial Approximation 1971-4c, 1984-4b, 1995-4a, 1995-4b, 1997-2d, 1998-3d Lagrange Error Bound 1976-7c, 1999-4b, 2004-6c Taylor or Maclaurin Series 1971-4a, 1982-5a, 1986-5a, 1990-5a, 1996-2a, 2003-6a, 2004-6b, 2005-6b Radius of Convergence 1984-4a, 2000-3b Infinite Series In Calculus Interval of Convergence 1971-4b, 1972-4b, 1975-4b, 1976-7b, 1979-4b, 1982-5b, 1983-5b, 1987-4b, 1988-4, 1991-5c, 1994-5b, 2001-6a, 2002-6a, 2005-6c Known Maclaurin Series 1976-7a, 1979-4a Manipulation of Known Series 1982-5d, 1986-5b, 1987-4a, 1987-4c, 1991-5a, 1993-5a, 1993-5b, 1994-5a, 1996-2b, 1996-2c, 1998-3b, 1999-4c, 2001-6b, 2003-6c, 2006-6a, 2006-6b, 2007-6a, 2007-6b Differentiation of Series 1983-5c, 1986-5c, 1993-5c, 1997-2b, 2002-6b, 2003-6c, 2006-6a Antidifferentiation of Series 1990-5b, 1991-5b, 1997-2c, 1998-3c, 2001-6c, 2004-6d, 2006-6c, 2007-6c The table above indicates that one of the most common types of questions involves the use of known series that are then modified through differentiation, integration, substitution (e.g., finding a Maclaurin series for sin(x 2) using the known series for sin(x)), and/or algebraic manipulation. These are all questions that manipulate the known series specified by the course description or manipulate a series given in the stem or another part of the problem. A second item commonly tested is that of interval of convergence. This most frequently involves using the ratio test to find the radius of convergence and then checking individually each of the endpoints of the interval generated by that radius of convergence to determine convergence or divergence. The endpoint checking has often involved the use of the alternating series test and knowledge of the convergence or divergence of well-known series, such as the harmonic series. The third most common question involves the use of the alternating series errorbound theorem. The typical form for this question is to use the first several terms of a Taylor polynomial to approximate the value of a function at a point and then either ask the question of how much error could have been made or have the student show that the error made in the approximation is less than a specified amount. In terms of the multiple-choice portion of the exam, the following types of questions were asked for the years when exams were released. 1997 Multiple-Choice Topics 1998 Multiple-Choice Topics 2003 Multiple-Choice Topics Sum of a Geometric Series Taylor Polynomial Approximation Geometric Series Sequence Limit Question Convergence Tests—Several Manipulation of Series Taylor Polynomial Integral Test Manipulation of Series for ex Interval of Convergence Differentiation of Series nth Term Test and p Series Differentiation of Series Geometric and Alternating Series Test Differentiation of Series and Taylor Series 7 Special Focus: Calculus Calculator—Error in Approximation Question Taylor Polynomials Interval of Convergence Exponential Series In regard to teaching infinite series, one thing is certain: They should be taught in the context of giving a different way of expressing a function, of their use as approximating (Taylor) polynomials, and of the error made in that approximation—whether it be given by the Lagrange error bound or by the alternating series error theorem. 8