RATIO MATHEMATICA ISSUE N. 30 206 pp. 59-66 ISSN print: 592-745 ISSN online: 2282-824 The sum of the series of reciprocals of the quadratic polynomial with different negative integer roots Radovan Potůček Department of Mathematics and Physics, Faculty of Military Technology, University of Defence, Brno, Czech Republic Radovan.Potucek@unob.cz Abstract This contribution, which is a follow-up to author s paper [] and [2] dealing with the sums of the series of reciprocals of some quadratic polynomials, deals with the series of reciprocals of the quadratic polynomials with different negative integer roots. We derive the formula for the sum of this series and verify it by some examples evaluated using the basic programming language of the CAS Maple 6. Keywords: sequence of partial sums, telescoping series, harmonic number, computer algebra system Maple. 200 AMS subject classifications: 40A05, 65B0. Introduction and basic notions Let us recall the basic terms, concepts and notions. For any sequence {a k } of numbers the associated series is defined as the sum a k = a + a 2 + a 3 +. k= 59
Radovan Potůček The sequence of partial sums {s n } associated to a series n as the sum of the sequence {a k } from a to a n, i.e. n s n = a k = a + a 2 + + a n. The series k= a k is defined for each a k converges to a limit s if and only if the sequence of partial sums k= k= {s n } converges to s, i.e. lim s n = s. We say that the series and write a k = s. k= a k has a sum s The telescoping series is any series where nearly every term cancels with a preceding or following term, so its partial sums eventually only have a fixed number of terms after cancellation. Telescoping series are not very common in mathematics but are interesting to study. The method of changing series whose terms are rational functions into telescoping series is that of transforming the rational functions by the method of partial fractions. For example, the series has the general nth term k 2 + k k= a n = nn + = A n + B n +. After removing the fractions we get the equation = An + + Bn. Solving it for A and B we obtain a n = /n /n +. After that we arrange the terms of the nth partial sum s n = a + a 2 + + a n in a form where can be seen what is cancelling. Then we find the limit of the sequence of the partial sums s n in order to find the sum s of the infinite telescoping series as s = lim s n. In our case we get s n = + 2 2 3 + + n n k= + n = n + n +. So we have s = lim =. n + The nth harmonic number is the sum of the reciprocals of the first n natural numbers: H n = + 2 + 3 + + n = 60 n k= k.
The sum of the series of reciprocals of the quadratic polynomial The values of the sequence {H n ln n} decrease monotonically towards the limit γ =. 0.5772566, which is so-called the Euler-Mascheroni constant. Basic information about harmonic numbers can be found e.g. in the web-sites [3] or [4], interesting information are included e.g. in the paper [5]. First ten values of the harmonic numbers are presented in this table: n 2 3 4 5 6 7 8 9 0 H n 3 2 6 25 2 37 60 49 20 363 40 76 280 729 2520 738 2520 2 The sum of the series of reciprocals of the quadratic polynomial with different negative integer roots Now, we deal with the series formed by reciprocals of the normalized quadratic polynomial k ak b, where a < b < 0 are integers. Let us consider the series k ak b, k= and determine its sum sa, b. We express the nth term a n of this series as the sum of two partial fractions a n = n an b = A n a + B n b. From the equality of two linear polynomials = An b + Bn a for n = a we get A = /a b and for n = b we get B = /b a = /a b. So we have a n = a b n a = n b b a n b. n a For the nth partial sum of the given series so we get s n = [ b a b + a 2 b + 2 a + n b + n a n b n a ]. The first terms that cancel each other will be obviously the terms for which for the suitable index l it holds / a = /l b. Therefore the last term from the beginning of the expression of the nth partial sum s n, which will not cancel, will 6
Radovan Potůček be the term / a, so that the first terms from the beginning of the expression the sum s n, which will not cancel, will be the terms generating the sum b + 2 b + + a. Analogously, the last terms that cancel each other will be the terms for which for the suitable index m it holds /n b = /m a. Therefore the first term from the ending of the expression of the nth partial sum s n, which will not cancel, will be the term /n + b, so that the last terms from the ending in the expression of the sum s n, which will not cancel, will be the terms generating the sum n + b n + 2 b n a. After cancelling all the inside terms with the opposite signs we get the nth partial sum s n = b a b + + + 2 b a n + b. n + 2 b n a Because for integer c it holds lim a < b < 0, is sa, b = lim s n = so we get n + c = 0, then the searched sum, where b a b + 2 b + + = a = [ b a + 2 + + a + 2 + + b ], Theorem 2.. The series, where a < b < 0 are integers, has k ak b k= the sum sa, b = H a H b, 2 b a where H n is the nth harmonic number. Corolary 2.. For the sum sa, b above it obviously hold:. sa, b = sb, a, 2. sa, a + = H a H a = a, 62
The sum of the series of reciprocals of the quadratic polynomial 3. sa, a+i = i i N. H a H a i = i a i + + + + a i + 2 a Remark 2.. Let us note, that the formula 2 holds also in the case a < b = 0. Because H 0 is defined as 0, it has the form Example 2.. The series k=, sa, 0 = H a H a H 0 = 0 a a. 3 k 5 k 2 = where a = 5, b = 2, has the nth partial sum s n = 3 k= k + 2k + 5, 3 + 4 + 5 n + 3 n + 4. n + 5 By the relation s 5, 2 = lim s n, since lim = 0, or by Theorem 2. n + c we get its sum s 5, 2 = 3 Example 2.2. The series 3 + 4 + = 37 H5 H 2 = 5 3 3 60 3 = 47 2 80 = 0.26. k= k 5 k = where a = 5, b = 0, has the nth partial sum s n = 5 k= kk + 5, + 2 + 3 + 4 + 5 n + n + 2 n + 3 n + 4. n + 5 By the relation s 5, 0 = lim s n, since lim = 0, or by Theorem 2., or n + c by the Remark 2. we get its sum s 5, 0 = 5 + 2 + 3 + 4 + = H 5 5 5 = 37/60 5 63 = 37 300 = 0.456.
Radovan Potůček 3 Numerical verification We solve the problem to determine the values of the sum sa, b = k= k ak b for a =, 2,..., 9 and b = a +, a + 2,..., 8. We use on the one hand an approximative direct evaluation of the sum sa, b, t = t k= k ak b, where t = 0 8, using the basic programming language of the computer algebra system Maple 6, and on the other hand the formula 2 for evaluation the sum sa, b. We compare 45 = 9 + 8 + + pairs of these ways obtained sums sa, b, 0 8 and sa, b to verify the formula 2. We use following simple procedures hnum, rp2abneg and two for statements: hnum:=proch local i,s; s:=0; if h=0 then s:=0 else for i from to h do s:=s+/i; end do; end if; end proc: rp2abneg:=proca,b,n local k,sab,sumab; sumab:=0; sab:=hnum-a-hnum-b/b-a; print"n=",n,"s",a,b,"=",evalf[20]sab; for k from to n do sumab:=sumab+/k-a*k-b; end do; print"sum",a,b,"=",evalf[20]sumab, "diff=",evalf[20]abssumab-sab; end proc: for i from - by - to -9 do for j from i+ by - to -8 do rp2abnegi,j,00000000; end do; end do; 64
The sum of the series of reciprocals of the quadratic polynomial The approximative values of the sums sa, b rounded to 3 decimals obtained by these procedures are written into the following table: sa, b a = a= 2 a= 3 a= 4 a= 5 a= 6 a= 7 a= 8 a= 9 b = 0.000 0.750 0.6 0.52 0.457 0.408 0.370 0.340 0.34 b = 0.500 0.47 0.36 0.32 0.290 0.266 0.245 0.229 b = 2 0.333 0.292 0.26 0.238 0.29 0.203 0.90 b = 3 0.250 0.225 0.206 0.90 0.77 0.66 b = 4 0.2000 0.83 0.70 0.59 0.49 b = 5 0.67 0.55 0.45 0.36 b = 6 0.43 0.34 0.26 b = 7 0.25 0.8 b = 8 0. Computation of 45 couples of the sums sa, b, 0 8 and sa, b took over 8 minutes. The absolute errors, i.e. the differences sa, b sa, b, 0 8, have here place value about 0 8. 4 Conclusion We dealt with the sum of the series of reciprocals of the quadratic polynomials with different negative integer roots a and b, i.e. with the series k= k ak b, where a < b < 0 are integers. We derived that the sum sa, b of this series is given by the formula sa, b = H a H b, b a where H n is the nth harmonic number. We verified this result by computing 45 various sums by using the CAS Maple 6. We also stated that this formula holds also for a < b = 0, when it has the form sa, 0 = H a H a H 0 = 0 a a. The series of reciprocals of the quadratic polynomials with different negative integer roots so belong to special types of infinite series, such as geometric and telescoping series, which sums are given analytically by means of a simple formula. 65
Radovan Potůček References [] R. Potůček, The sums of the series of reciprocals of some quadratic polynomials. In: Proceedings of AFASES 200, 2th International Conference Scientific Research and Education in the Air Force CD-ROM. Brasov, Romania, 200, p. 206-209. ISBN 978-973-845-76-8. [2] R. Potůček, The sum of the series of reciprocals of the quadratic polynomials with double non-positive integer root. In: Proceedings of the 5th Conference on Applied Mathematics APLIMAT 206. Faculty of Mechanical Engineering, Slovak University of Technology in Bratislava, 206, p. 99-925. ISBN 978-80-227-453-4. [3] Wikipedia contributors: Harmonic number. Wikipedia, The Free Encyclopedia, [online], [cit. 206-09-0]. Available from: https://en.wikipedia.org/wiki/harmonic number. [4] E. W. Weisstein, Harmonic Number. From MathWorld A Wolfram Web Resource, [online], [cit. 206-09-0]. Available from: http://mathworld.wolfram.com/harmonicnumber.html [5] A. T. Benjamin, G. O. Preston, and J. J. Quinn, A Stirling Encounter with Harmonic Numbers. Mathematics Magazine 75 2, 2002, p. 95 03, [online], [cit. 206-09-0]. Available from: https://www.math.hmc.edu/ benjamin/papers/harmonic.pdf 66