Divergent Series wear White Hats, too

Transcription

1 Divergent Series wear White Hats, too J. B. Thoo 1 Department of Mathematics University of California Davis, CA USA jb2@math.ucdavis.edu October 6, Student.

2 Abstract Using divergent series, one can obtain reasonably accurate numerical approximations of function values with minimal work; unfortunately, divergent series are downplayed in the usual freshman calculus sequence. In this paper, we explain why divergent series are useful, and give two examples of their use.

3 1 Introduction Divergent series are not given their due attention in the usual freshman calculus sequence. Certainly when I learned about infinite series for the first time, I was left with the distinct impression that absolutely convergent series were best, convergent series were good, and divergent series were something to avoid. Nothing I was taught or learned following that warranted a change of heart in the matter; indeed, if anything, I became surer of this belief than ever before. Hence, you can imagine my amazement when, in a recent asymptotic analysis course, I learned that divergent series were not only useful, but that in many cases were of more practical use than convergent series. How is it that a divergent series could be of any practical value? The reason is really quite simple: even though a power series might converge (pointwise) to a function, it might require an enormous number of terms before the error is acceptable; a divergent series, on the other hand, might approximate a function value quite well in a small number of terms before it deviates from it drastically. 1 1 More precisely, we call the series a kφ k(t) anasymptotic approximation of the function f(t) ast a if for each n 1. φ n+1(t) =o(φ n(t)) as t a (i.e. {φ k(t)} is an asymptotic sequence as t a), and 1

4 2 A first example A favorite among elementary calculus textbooks is using the series expansion e x2 = k= ( x 2 ) k /k! toevaluate e x2 dx. (1) This is usually touted as a great triumph because, as the student is reminded, the function e x2 does not have an elementary antiderivative. For instance, Barcellos and Stein [1] give the example = k= k= ( x 2 ) k k! ( x 2 ) k k! dx dx = k= ( 1) k t 2k+1 (2k +1)k! (2) with t = 1. As it turns out, only five terms of (2) are required for the approximation to agree to three decimal places with the result 1 e x2 dx = obtained via numerical integration. 2 Unfortunately, this partic- 2. f(t) = n akφk(t)+o(φn+1(t)) as t a (i.e. the error is of the same order of k=1 magnitude as the first omitted term). It is this asymptoticness that provides for a good approximation using only a few terms of the series. Note that a series convergent or divergent need not be asymptotic as t a. For a further discussion of asymptotic series, see e.g. [4, 5]; also see [2, 3] for a brief history, as well as a selection of historical examples. 2 Computations were obtained using The Student Edition of Theorist R by Prescience Corporation and PWS Publishing Company. 2

5 ular example is somewhat misleading because, even though the series (2) converges for all real t, it is very good for approximating (1) only when t is small. For example, to agree to three decimal places with the numerical result (3) requires twenty-six terms of (2); to obtain similar accuracy with 5 e x2 dx = requires sixty-eight terms; and the number of terms required to maintain that accuracy continues to grow dramatically as one integrates over an increasingly longer interval [,t]. As Hinch [4] suggests, the way to evaluate (1) for large values of t is to use the alternative representation π 2 e x2 dx. (4) t The key is to observe that repeated integration by parts in the right-hand side of (4) gives ( ) 1 d t t 2x dx e x2 dx ( ) = e t2 1 2t t 2x 2 e x2 dx. = e t2 2t ( 1 k= ) ( 1) k (2k +1) (2t 2 ) k+1, 3

6 which yields formally ( π 2 e t2 1 2t k= ) ( 1) k (2k +1) (2t 2 ) k+1. (5) The surprise here is that even though the series (5) diverges, using only two terms yields a result that agrees to three decimal places with (3). Indeed, following Hinch, we see that because π 2 e t2 2t + R, where ( ) R = 1 ( ) t 2x 2 e x2 dx 1 = d t 4x 3 dx e x2 dx < 1 d 4t 3 t dx e x2 dx (since x 3 <t 3 for t<x) = e t2 4t 3, only two terms of (5) are necessary to obtain an accuracy of 1 5 when t 3 truly remarkable! It should be noted, however, that because the series (5) diverges, one does not improve the accuracy of the approximation by using increasingly many terms of the series. For example, using thirty terms of (5) yields a result that agrees with (3) to only two decimal places (down from the threedecimal-place accuracy obtained using only two terms), and increasing the 4

7 number of terms beyond that causes the approximation to deviate wildly fromthetruevalue. The reason that truncating the divergent series (5) provides us with a useful approximation of (1) for large values of t is that the series is asymptotic to the function f(t) = as t so that e x2 dx (6) ( π n 1 2 e t2 1 2t k= ( 1) k ) (2k +1) (2t 2 ) k+1 +O (t 2(n+1)) as t ; by contrast, the series (2) is asymptotic to the function (6) as t. The latter explains why (2) provides a good approximation of (1) for small values of t. 3 Another example For another example, 3 consider the function y(t) = Observe that (we omit the details) e x dx. (7) 1+tx 3 HomeworkprobleminJ.K.Hunter sasymptotic Analysis course, UC Davis, Fall 92. 5

8 1. y(t) is defined for t (it has a nonintegrable singularity at x = 1/t for t<). 2. By repeatedly integrating (7) by parts, one has formally that y(t) = ( 1) k k!t k. (8) k= 3. The series (8) diverges for all t. 4. Numerical integration of (7) gives y(.1) = Computing the partial sums n S n (t) = ( 1) k k!t k k= of (8) for t =.1andn =1, 5, 1, 15, 2, one finds that the partial sum S 1 gives the best agreement (to three decimal places). 5. One has the error estimate e x 1+tx dx S n(t) (n +1)!tn For fixed t, the error term is smallest when (n +1)!t n+1 n!t n and (n +1)!t n+1 < (n +2)!t n+2, i.e. at n = 1/t 1, where t denotes the integer part of t. (Thus,fort =.1 theoptimal approximation is in fact given by S 9.) 6

9 7. Using Stirling s formula, one can deduce further that the error of the optimal approximation is exponentially small as t +. 4 Conclusion Many perhaps most of our students taking the freshman calculus sequence for scientists and engineers are, in fact, engineering majors. For this reason alone, we should seriously consider shedding better light on divergent series, for they are proven tools in obtaining reasonably accurate numerical results with minimal work. At the very least, we should tell our students that divergent series wear white hats, too. References [1] Barcellos, A., andstein, S. K., 1992, Calculus and Analytic Geometry, fifth edition (New York: McGraw-Hill). [2] Ford,W.B., 196, Studies on Divergent Series and Summability, and the Asymptotic Developments of Functions Defined by Maclaurin Series (New York: Chelsea Publishing Company). 7

10 [3] Hardy, G. H., 1991, Divergent Series, second edition, textually unaltered (New York: Chelsea Publishing Company). [4] Hinch, E. J., 1991, Perturbation Methods (New York: Cambridge University Press). [5] Murray, J. D., 1984, Asymptotic Analysis (New York: Springer- Verlag). 8