Inequalities for finite trigonometric sums. An interplay: with some series related to harmonic numbers

Transcription

1 Kouba Journal of Inequalities and Alications 6 6:73 DOI.86/s R E S E A R C H Oen Access Inequalities for finite trigonometric sums. An interlay: with some series related to harmonic numbers Omran Kouba * * Corresondence: omran_kouba@hiast.edu.sy Deartment of Mathematics Higher Institute for Alied Sciences and Technology P.O. Box 3983 Damascus Syria Abstract An interlay between the sum of certain series related to harmonic numbers and certain finite trigonometric sums is investigated. This allows us to exress the sum of these series in terms of the considered trigonometric sums and ermits us to find shar inequalities bounding these trigonometric sums. In articular this answers ositively an oen roblem of Chen Excursions in Classical Analysis. MSC: B68; B83; 6D5; 6D5; 4A7 Keywords: Bernoulli olynomials; Bernoulli numbers; harmonic numbers; asymtotic exansion; sum of cosecants; sum of cotangents Introduction Many identities that evaluate trigonometric sums in closed form can be found in the literature. For examle in a solution to a roblem in SIAM Review [].57 Fisher shows that k sec = 3 k sec 4 = General results giving closed forms for the ower sums of secants secn k and secn k for many values of the ositive integer n canbefoundin[3] and[4]. + Also in [5] the author roves that k sec + if is even = if is odd. However while there are many cases where closed forms for finite trigonometric sums can be obtained it seems that there are no such formulas for the sums we are interested in. 6 Kouba. This article is distributed under the terms of the Creative Commons Attribution 4. International License htt://creativecommons.org/licenses/by/4./ which ermits unrestricted use distribution and reroduction in any medium rovided you give aroriate credit to the original authors and the source rovide a link to the Creative Commons license and indicate if changes were made.

2 Kouba Journal of Inequalities and Alications 6 6:73 Page of 5 In this aer we study the trigonometric sums I and J defined for ositive integers by the formulas I = sink/ = k csc k J = k cot.. with emty sums interreted as. To the best of the author s knowledge there no closed form for I is known and the same can be said about the sum J. Therefore we will look for asymtotic exansions for these sums and will give some tight inequalities that bound I and J. This investigation comlements the work of Chen in []Chater7whereitwasaskedasanoenroblem whether the inequality I ln + γ ln/ holds true for 3hereγ is the so-called the Euler-Mascheroni constant. In fact it will be roved that for every ositive integer and every nonnegative integer nwehave and I < I > ln + γ ln/ + n n+ ln + γ ln/ + k k b k k k! k k b k k k! k k where the b k s are Bernoulli numbers see Theorem 3.4. The corresonding inequalities for J are also roved see Theorem 3.9. Harmonic numbers lay an imortant role in this investigation. Recall that the nth harmonic number H n is defined by H n = n /k with the convention H =.Inthiswork a link between our trigonometric sums I and J and the sum of several series related to harmonic numbers is uncovered. Indeed the well-known fact that H n = ln n + γ + n + O roves the convergence of the numerical series n C = D = E = H n lnn γ n n H n lnn γ n H n+ H n n= for every ositive integer.

3 Kouba Journal of Inequalities and Alications 6 6:73 Page 3 of 5 An interlay between the considered trigonometric sums and the sum of these series will allow us to rove shar inequalities for I and J and to find the exression of the sums C D ande in terms of I and J. Themaintoolwillbethefollowingformulation[6] Corollary 8. of the Euler- Maclaurin summation formula. Theorem. Consider a ositive integer m and a function f that has a continuous m st derivative on [ ]. If f m is decreasing then f t dt = f + f m b k k! δf k + m+ R m with and / R m = B m t f m t f m t dt m! R m 6 m f m f m where the b k s are Bernoulli numbers B m is the Bernoulli olynomial of degree m and the notation δg for a function g :[] C means g g. For more information on the Bernoulli olynomials Bernoulli numbers and the Euler- Maclaurin formula the reader may refer to [6 ] and the references therein. This aer is organized as follows. In Section we find the asymtotic exansions of C and D for large.insection3 the inequalities the trigonometric sums I and J are roved. Asymtotic exansions for the sum of certain series related to harmonic numbers In the next lemma the asymtotic exansion of H n n N is resented. It can be found imlicitly in Chater 9 of []; we resent a roof for convenience of the reader. Lemma. For every ositive integer n and nonnegative integer m we have with H n = ln n + γ + m n R nm = / B m t b k k n k + m R nm j + t m j + t m dt. Moreover<R nm < b m m n m.

4 Kouba Journal of Inequalities and Alications 6 6:73 Page 4 of 5 Proof Note that for j wehave j ln + = j j j + t dt = t jj + t dt. Adding these equalities as j varies from to n weconcludethat H n ln n n n = t dt. jj + t j= Thus letting n tend to and using the monotone convergence theorem we conclude t γ = dt. jj + t j= It follows that γ + ln n H n + n = t dt. jj + t So let us consider the function f n :[] R defined by f n t= t jj + t. Note that f n = f n = /nandthatf n is infinitely continuously derivable with f n k t = k+ fork. k! j + t k+ In articular f n k t k! = fork. j + t k So f m n is decreasing on the interval [ ] and δf n k k! = j + k = jk n. k Alying Theorem. to f n and using the above data we get with γ + ln n H n + m n = b k kn k + m+ R nm R nm = / B m t j + t m j + t m dt

5 Kouba Journal of Inequalities and Alications 6 6:73 Page 5 of 5 and <R nm < 6 m! m n m. The imortant estimate is the lower bound i.e. R nm >. In fact considering searately the cases m odd and m even we obtain for every nonnegative integer m : and H n < ln n + γ + m n H n > ln n + γ + m + n b k k b k k n k n k. This yields the following more recise estimate for the error term: < m H n ln n γ m n + b k k n k < b m m n m which is valid for every ositive integer m. Now consider the two sequences c n n and d n n defined by c n = H n ln n γ n and d n = H n ln n γ. For a ositive integer we know according to Lemma. that c n = O it follows n that the series c n is convergent. Similarly since d n = c n + and the series n n /n isconvergentweconcludethat n d n is also convergent. In the following we aim to find asymtotic exansions for large of the following sums: C = c n = H n lnn γ n. D = n d n = n H n lnn γ. E = n H n+ H n..3 n= Theorem. If and m are ositive integers and C is defined by. then m C = b k ζ k k k + m ζ m m m ε m with <ε m < b m where ζ is the well-known Riemann zeta function.

6 Kouba Journal of Inequalities and Alications 6 6:73 Page 6 of 5 Proof Indeed we conclude from Lemma. that H n lnn γ m n = with < r nm b m. It follows that m C = b k k k where r m = r nm. n m Hence < r m = r nm n m < b m b k k k n k + + m n k m r m m n m = b m ζ m m rnm m m n m and the desired conclusion follows with ε m = r m /ζ m. For examle taking m =3weobtain H n lnn γ n = O. 4 6 In the next roosition we have the analogous result corresonding to D. Theorem.3 If and m are ositive integers and D is defined by. then D = ln m b k ηk ηm + m k k m m ε m with <ε m < b m where η is the Dirichlet eta function []. Proof Indeed let us define a nm by the formula a nm = H n ln n γ m n + b k k n k with emty sum equal to. We have shown in the roof of Lemma. that m a nm = / Bm t gnm t dt where g nm is the ositive decreasing function on [ /] defined by g nm t= j + t m j + t m.

7 Kouba Journal of Inequalities and Alications 6 6:73 Page 7 of 5 Now for every t [ /] the sequence g nm t n is ositive and decreasing to. So using the alternating series criterion [3] Theorem7.8 and Corollary7.9 we see that for every N andt [ /] n g nm t g Nmt g Nm = N. m n=n This roves the uniform convergence on [ /] of the series G m t= Consequently n g nm t. m n a nm = / Bm t Gm t dt. Now using the roerties of an alternating series we see that for t / we have <G m t<g m t<g m = j= j = m j + m. m Thus n a nm = m m ρ m with < ρ m < / B m t dt. On the other hand we have Thus n a nm = D D = ln m n = D ln m + b k ηk k k n + m ρ m m m + b k ηk k k. b k k k n Now the imortant estimate for ρ m is the lower bound i.e. ρ m >. In fact considering searately the cases m odd and m even we obtain for every nonnegative integer m : n k D < ln m b k ηk k k

8 Kouba Journal of Inequalities and Alications 6 6:73 Page 8 of 5 and D > ln m + b k ηk k k. This yields the following more recise estimate for the error term: < m D ln m + b k ηk < b m ηm k k m m and the desired conclusion follows. ThecaseofE which is the sum of another alternating series.3 is discussed in the next lemma where it is shown that E can easily be exressed in terms of D. Lemma.4 For a ositive integer we have E = ln + γ ln +D where D is the sum defined by.. Proof Indeed D = d + n d n + n d n n= = d + n d n+ + n d n = d + n d n d n+ n + = d + n H n+ H n + n ln n n + = ln γ + n H n+ H n + n ln. n n= Using the Wallis formula for [8] Formula.6 we have n + n ln = n n n ln n n + = ln 4n = ln and the desired formula follows.

9 Kouba Journal of Inequalities and Alications 6 6:73 Page 9 of 5 3 Inequalities for trigonometric sums As we mentioned in the introduction we are interested in the sum of cosecants I defined by. and the sum of cotangents J defined by.. Many other trigonometric sums can be exressed in terms of I and J. The next lemma lists some of these identities. Lemma 3. For a ositive integer let k k K = tan K = cot k L = sink/ M = Then: i K = K = I. ii L =/I. iii M =/J J = I. k k cot Proof First note that the change of summation variable k k roves that K = K. So using the trigonometric identity tanθ + cot θ = cscθ we obtain ias follows: k K = K + K = tan + cot. k k = csc =I. Similarly ii follows from the change of summation variable k k in L : Also But k L = sink/ = I L. M = k< k odd = k k cot k J = k cot = k k cot k k k cot k cot + k=+ k k cot k k = k cot k cot k = k cot K. k< k even = J J. k k cot

10 Kouba Journal of Inequalities and Alications 6 6:73 Page of 5 Thus using i and the trigonometric identity cotθ/ cot θ = cscθ we obtain M = J J = k cot k = k csc k cot k I I =L I = I. This concludes the roof of iii. Proosition 3. For let I be the sum of cosecants defined by the.. Then I = ln = ln + E + ln + γ ln/ + 4 D where D and E are defined by Eqs.. and.3 resectively. Proof Indeed our starting oint will be the simle fraction exansion [4] Chater 5 Section of the cosecant function: sinα = n α n = α + n α n + α + n n Z which is valid for α C \ Z. Usingthisformulawithα = k/ for k =... and adding we conclude that I = k + n = k + n k n + k + n n + n+ j + j + and this result can be exressed in terms of the harmonic numbers as follows: I = H + n H n + H n + H n+ H n = H + n H n+ H n + H n + n n n + = H + n H n+ H n + H n + n n + n n = H + n H n+ H n + H n n n = H ln + n H n+ H n + H n. n=

11 Kouba Journal of Inequalities and Alications 6 6:73 Page of 5 Thus I + ln = H + n= n H n+ H n + n H n H n = n H n+ H n + n H n H n = E + E =E and the desired formula follows according to Lemma.4. Combining Proosition 3. and Theorem.3 we obtain thefollowing. Proosition 3.3 For and m we have I =ln + γ ln/ m with <ε m < b m. b k ηk k k Using the well-known result [8] [] Formula 9.54: + m ηm m m ε m ηk= k ζ k= k k k b k k! and considering searately the cases m even and m odd we obtain the following result. Theorem 3.4 For every ositive integer and every nonnegative integer n the sumof cosecants I defined by. satisfies the following inequalities: I < I > ln + γ ln/ + n n+ ln + γ ln/ + k k b k k k! k k b k k k! k k. As an examle for n = we obtain the following inequality valid for every : ln + γ ln/ 36 < I < ln + γ ln/. This answers ositively the oen roblem roosed in Section 7.4 of []. Remark 3.5 The asymtotic exansion of I wasroosedasanexercisein[9] Exercise3.46anditwasattributedtoWaldvogelbuttheresultthereislessrecisethan Theorem 3.4 because here we have inequalities valid in the whole range of. Now we turn our attention to the other trigonometric sum J. The first ste is to find an analogous result to Proosition 3. for the trigonometric sum J isthenextlemmawhere

12 Kouba Journal of Inequalities and Alications 6 6:73 Page of 5 an asymtotic exansion for J is roved but it has a harmonic number as an undesired term; later it will be removed. Lemma 3.6 For every ositive integers there is a real number θ such that J = H + ln θ. Proof Indeed let ϕ be the function defined by ϕx=xcotx+ x. According to the artial fraction exansion formula for the cotangent function [4] Chater 5 Section we know that ϕx=+ x x + + x n= x n + x. x + n Thus ϕ is defined and analytic on the interval. Let us show that ϕ is concave on this interval. Indeed it is straightforward to check that for < x <wehave ϕ x= + x 3 n= n n x 3 + n n + x 3 <. SowecanuseTheorem. with m = alied to the function x ϕ x+k for k < to get < k+/ k/ ϕx dx ϕ k + + ϕ k 3 k k + ϕ ϕ. Adding these inequalities and noting that ϕ = ϕ = ϕ = and ϕ = /3 we get < ϕx dx J H 3+ <. Also for x [ we have x ϕt dt = ln x +x ln sinx and letting x tend to we obtain ϕt dt = ln x ln sint dt = ln ln sint dt whereweusedthefact ln sint dt = ln see[8] 4.4 Formula 3. So we have roved that < ln J H < which is equivalent to the desired conclusion.

13 Kouba Journal of Inequalities and Alications 6 6:73 Page 3 of 5 The next roosition gives an analogous result to Proosition 3. for the trigonometric sum J. Proosition 3.7 For a ositive integer let J be the sum of cotangents defined by.. Then J = ln + ln γ + C where C is given by.. Proof Recall that c n = H n ln n γ n satisfies c n = O/n. Thus the two series C = c n and C = n c n are convergent. Further we note that C = D ln where D is defined by.. According to Proosition 3. we have C = Now noting that ln/ γ ln C = c n + c n = c n + n n n n odd n even n odd C = c n c n = c n n n n n odd n even n odd + 4 I. 3. c n c n we conclude that C C =C orequivalently C C = C. 3. On the other hand for a ositive integer let us define F by F = ln + γ ln It is easy to check using Lemma 3.iii that + + J. 3.3 F F = = ln/ ln γ ln/ ln γ 4 J J + 4 I. 3.4 We conclude from 3.and3.4thatC C = F F orequivalently C F =C F.

14 Kouba Journal of Inequalities and Alications 6 6:73 Page 4 of 5 Hence m C F = m C m F m. 3.5 Now using Lemma. to relace H in Lemma 3.6weobtain J = ln H + O = ln ln γ + O. Thus F = O. Similarly from the fact that c n = O weconcludealsothatc n = O. Consequently there exists a constant κ such that for large values of wehave C F κ/. So from 3.5 we see that for large values of m we have C F κ m and letting m tend to + we obtain C = F which is equivalent to the announced result. Combining Proosition 3.7 and Theorem. we obtain the following. Proosition 3.8 For and m we have J = ln + ln γ m b k ζ k ζ m + m k k m ε m m with <ε m < b m where ζ is the well-known Riemann zeta function. Using the values of the ζ k s [8] Formula 9.54 and considering searately the cases m even and m odd we obtain the next result. Theorem 3.9 For every ositive integer and every nonnegative integer n the sum of cotangents J defined by. satisfies the following inequalities: J < J > ln + ln γ + n ln + ln γ n+ + k b k k k k! k b k k k! k. As an examle for n = we obtain the following double inequality which is valid for : < ln + ln γ J < 36.

15 Kouba Journal of Inequalities and Alications 6 6:73 Page 5 of 5 Remark 3. Note that we have roved the following results. For a ositive integer : n H n lnn γ = n= n H n+ H n = ln H n lnn γ n ln/ γ ln + k csc + ln + 4 ln + γ ln = + + These results are to be comared with those in [5]; see also [6]. k csc k k cot. Cometing interests The author declares that there are no cometing interests with any individual or institution and that he has not received any financial suort to do this research. Author s contributions The author declares that this work was carried out by himself. Acknowledgements The author would like to thank the anonymous referees for reading this article carefully and roviding valuable suggestions. Received: 7 March 6 Acceted: June 6 References. Chen H: Excursions in Classical Analysis. Math. Assoc. of America Washington. Klamkin MS: Problems in Alied Mathematics: Selections from SIAM Review. SIAM Philadelhia 99. doi:.37/ Chen H: On some trigonometric ower sums. Int. J. Math. Math. Sci Grabner PJ Prodinger H: Secant and cosecant sums and Bernoulli-Nörlund olynomials. Quaest. Math Kouba O Andreescu T: Mathematical Reflections Two More Years - Solution to Problem U7. XYZ Press San Jose 4 6. Kouba O: Lecture notes Bernoulli olynomials and alications v 7. Abramowitz M Stegan IA: Handbook of Mathematical Functions: With Formulas Grahs and Mathematical Tables. Dover Books on Mathematics. Dover New York Gradshteyn I Ryzhik I: Tables of Integrals Series and Products 7th edn. Academic Press San Diego 7 9. Henrici P: Alied and Comutational Comlex Analysis vol.. Wiley New York 977. Olver FWJ: Asymtotics and Secial Functions. Academic Press New York 974. Graham RL Knuth DE Patashnik O: Concrete Mathematics: A Foundation for Comuter Science nd edn. Addison-Wesley Reading 994. Weisstein EW: Dirichlet eta function. From MathWorld - A Wolfram web resource. htt://mathworld.wolfram.com/dirichletetafunction.html 3. Amann H Escher J: Analysis I. Birkhäuser Basel 5 4. Ahlfors LV: Comlex Analysis. McGraw-Hill New York Kouba O: The sum of certain series related to harmonic numbers. Octogon Math. Mag Kouba O: Proosed Problem 499. Am. Math. Mon