Power Series for Practical Purposes Ralph Boas The Two-Year College Mathematics Journal, Vol. 13, No. 3. (Jun., 1982), pp. 191-195. Stable URL: http://links.jstor.org/sici?sici=0049-4925%28198206%2913%3A3%3C191%3APSFPP%3E2.0.CO%3B2-0 The Two-Year College Mathematics Journal is currently published by Mathematical Association of America. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/maa.html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academic journals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers, and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community take advantage of advances in technology. For more information regarding JSTOR, please contact [email protected] http://www.jstor.org Wed Aug 22 11:14:22 2007 Power Series for Practical Purposes "Take a bone from a dog-what remains?"-The White Queen Ralph Boas The author retired from Northwestern University in 1980 with the title of Professor Emeritus of Mathematics, a title which, he likes to point out, has nothing to do with merit. He has also retired as editor of the "American Mathematical Monthly;" the December 1981 issue has a symbolic portrait of him on the cover. This article is an outgrowth of his hobby of trying to encourage people to look at apparently complicated things in a simple way. 1. Why Taylor series? We hear a great deal about teaching mathematics that is applicable to "the real world." Consequently I was nonplussed at being told recently, in all seriousness, that most teachers will reject a calculus text out of hand if it doesn't discuss the formula for the remainder in a Taylor series. This claim may well be true; but, if so, most teachers are trying to teach the wrong course. The Taylor remainder is an important piece of mathematics associated with calculus-important for proving theorems, that is. So are uniform convergence and Lebesgue integration, but most teachers realize that these topics will not be appreciated by the mass of freshmen and sophomores. Why is the Taylor remainder conceived to be of such central importance? The conventional answer is that with the remainder formula we can estimate remainders and so be sure that power series represent the functions we got them from. This is indeed true for the exponential, sine and cosine functions, for the binomial series-and not for much of anything else. Indeed, these are practically the only elementary functions whose successive derivatives are simple enough to calculate in the first place and then to estimate. Just try to calculate enough derivatives to estimate the remainder after eight terms for cos(x - sinx). (This function occurs in the theory of frequency modulation.) You could, of course, use a computer to evaluate the derivatives, but in that case you probably wouldn't need the series anyway. Given that the remainder formula is useful primarily for theoretical purposes, can we justify the introduction of Taylor series at all? Isn't it now only of academic interest that Not long ago, if you wanted a fairly precise approximation to e0.0'982, you couldn't do much better than to calculate a few terms of the series with x = 0.01982. Nowadays, of course, you push seven buttons on the little pocket calculator and get 1.02001772, a result more precise than anything you are likely to need. You can obtain values of cos(x - sinx) almost as easily. Nevertheless, many people who apply mathematics do value the ability to write down several terms of the series of such functions. The usual technique for cos(x - sinx) would be to form the Maclaurin series for x - sinx, substitute this series for y in the Maclaurin series for cosy, and rearrange the result as a power series in x. Authors of calculus books seem to be largely unaware that this is a completely legitimate process. But, given the existence of calculators and computers, why should anyone bother to work out the Maclaurin series by any method? In the first place there are things that calculators don't yet do, or do only after special programming. One of these is the evaluation of nonelementary functions. I have seen a calculator that has built-in gamma functions, but, as far as I know, there is as yet no calculator with built-in Bessel functions or elliptic integrals. Another operation that is not easy on a calculator is the calculation of a table of values of an indefinite integral. Still another is making calculations of higher precision than was built in. You cannot, at least at present, appeal directly to a calculator for values of Here J , is the Bessel function (2) is an elliptic integral, and (3) is a solution of [ l ] x?" + x?' + (x2 - 2)y = 0. It is relatively easy to start from well-known power series and work out the first few coefficients of the Maclaurin series of (I), (2) or (3). If one of these functions turns up in a practical problem, it will probably have to be entered into a computer to be used for further computation. It is much easier to enter a small number of coefficients than a whole table of values. 2. Why It works. To get power series for functions like (I), (2) or (3), we need to know that power series can be differentiated or integrated, multiplied or divided, and (most importantly) substituted into each other; then, if necessary, rearranged as power series again, with the same result as if we had worked out the coefficients by differentiation. Some of the proofs are beyond the scope of a calculus course, but at least correct statements of the relevant theorems are easily understood. (The same can be said for other theorems that are used in calculus courses.) It is most convenient to state the theorems for complex series; but they can be specialized (although not all the proofs can) to real series by reading "interval of convergence" for "disk of convergence." A. Differentiation and integration are permissible in the interior of the disk of convergence. This is straightforward to prove and is done in many textbooks. B. Multiplication. If the Maclaurin series of f and g both converge for then their formal product, arranged as a series of ascending powers, has radius of convergence at least r and represents fg. (Here r is not necessarily the radius of convergence of either series.) This is a special case of the theorem that the formal product of two absolutely convergent series converges (absolutely) to the product of the sums of the series. In fact, we have IzI < r, x 00 n=O x 00 a,zn. x 00 k bmzm= ckzk, where ck = ajbk_,. m=O k=O j=O Hence we can calculate the coefficients in the product series or program them for a computer to calculate. C. Division. If the Maclaurin series of f and g converge for lzl < r and g(z) 0 for 0 < lzl < r, then if the Maclaurin series for f is divided by the Maclaurin series for g by long division (as if the series were polynomials), the resulting series represents f / g for lzl < r. Again, r is not necessarily the radius of convergence of either series. If h = f / g then f = gh. Write h(z) = C;='=,cnzn,multiply h by g by using the formula under B, and solve for the cn recursively. We can then see by induction that these c, are exactly what one gets by long division. For example, C does not allow us to divide the series for cosz by the series for sinz near z = 0; fortunately, since cotz does not have a Maclaurin series. + D. Substitution. The following theorem, although not the most general possible, covers many cases that are likely to arise in practice. Curiously enough, it is absent from most calculus books and is not discussed adequately in most modern textbooks on complex analysis. It was rather difficult to locate a formal proof (see, however,  or ); I give one in 03 in case anyone wants to see it. It involves nothing deeper than the theorem that a uniformly convergent series of analytic functions can be differentiated term by term. Substitution theorem. Let f(w) be represented by the series C;='=,anwnfor lzl < r, and let I g(0)I < s. Then the Maclaurin series of F(z) = f(g(z)) can be obtained by substituting w = C?=',,bkzk into C;='=,a,wn and rearranging the result as a power series in t. IwI < S; let g(z) be represented by C:='=,bkzk for The resulting series represents F(z) in any disk lzl < t in which F is analytic; there is such a disk because F is analytic at 0. In practice, f and g are often elementary functions and the radius of convergence of the Maclaurin series of F can be found by inspection. Notice particularly what the theorem does not permit. For example, we cannot get a Maclaurin series for ln(1 - cosz) by substituting the Maclaurin series for cos z into the series for ln(1 - w), because the composite function is not even defined at z = 0. The theorem does not let us try because the Maclaurin series for ln(1 - w) has radius of convergence s = 1 but cosO = 1 is not less than s. We must also be careful to avoid the mistake of substituting the Maclaurin series of g into the Maclaurin series of f when g is analytic at z = a but g(a) is outside the disk of convergence of the Maclaurin series of f. For example, consider a branch of (1 - 2 c o ~ z ) ' / ~The . Maclaurin series of (1 - 2w)'l2 converges only for Iwl < f , but cosO = 1; even though (1 - 2 c 0 s z ) ' / ~ is analytic at z = 1, we cannot substitute w = cosz into a divergent Maclaurin series and expect to get a meaningful result when z is near 0. 3. Proof of the substitution theorem. Since F(z) is ,analytic at 0 it has a Maclaurin series C h n z n .On the other hand, when I g(z)l < s we have by the multiplication theorem. What we want to show is that Now CA,zn is uniformly convergent in any closed disk lzl < r where F is analytic and hence can be differentiated repeatedly in a neighborhood of 0. Setting z = 0, we get Now C+,wm converges uniformly for Iwl < s , < s and so C+,[g(z)lm converges uniformly provided I g(z)l < s l . This will be the case if lzl is sufficiently small and s , is sufficiently close to s, since I g(0)I < s. Hence C+,[g(z)lm, although not a power series, can also be differentiated term by term; consequently setting z =0 we get This process can be continued to yield (5). This shows that the series for f(g(z)) can be rearranged into a power series when z is sufficiently close to 0; this series then represents F(z) in the largest disk, center at 0, in which F is analytic. REFERENCES I. E. Kamke, Differentialgleichungen, Losungsmethoden und Losungen, vol. 1, 3rd ed., Leipzig, Akademische Verlagsgesellschaft, 1944, p. 450, (2.208). 2. A. I. Markushevich, Theory of Functions of a Complex Variable, vol. I (translated and edited by R. A. Silverman), Prentice-Hall, Englewood Cliffs, N.J., 1965, p. 433. 3. W. F. Osgood, Lehrbuch der Funktionentheorie, vol. 1, 5th ed., Teubner, Leipzig, 1928, p. 362. CLASSROOM CAPSULES Edited by Warren Page Classroom Capsules serves to convey new insights on familiar topics and to enhance pedagogy through shared teaching experiences. Its format consists primarily of readily understood mathematics capsules which make their impact quickly and effectively. Such tidbits should be nurtured, cultivated, and presented for the benefit of your colleagues elsewhere. Queries, when available, will round out the column and serve to open further dialog on specific items of reader concern. Readers are invited to submit material for consideration to: Warren Page Department of Mathematics New York City Technical College 300 Jay Street Brooklyn, N. Y. 11201 Intuition Out to Sea William A. Leonard, California State University, Fullerton, CA Using only intuition and without writing anything, try to answer the following question: A rope, attached to boat B and passing over pulley C, is drawn to the right one foot at A . Does the boat move more than one foot, less than one foot, or exactly one foot to the right?