ON THE CONVERGENCE OF SOME ALTERNATING SERIES

Transcription

1 O THE COVERGECE OF SOME ALTERATIG SERIES AGEL V. KUMCHEV. Introduction This note is motivated by a question that a colleague of the author s often challenges calculus students with: Does the series ( ) n sin n () n converge? This series combines features of several series commonly studied in calculus: ( ) n n, sin n sin(nx) and n n come to mind. However, unlike these familiar examples, the series () seems to live on the fringes, just beyond the reach of standard convergence tests like the alternating series test or the tests of Abel and Dirichlet. It is therefore tempting for an infinite series aficionado to study () in hope to find some clever resolution of the question of its convergence. Yet, the author s colleague reports that although he has posed the above question to many calculus students, he has never received an answer. It turns out that there is a good reason for that: the question is quite delicate and is intimately connected to deep facts about Diophantine approximation facts which the typical second-semester calculus student is unlikely to know. The series () is obtained by perturbation of the moduli of the alternating harmonic series, which is the simplest conditionally convergent alternating series one can imagine. In this note, we study the convergence sets of similar perturbations of a wide class of alternating series. In particular, the convergence of () follows from our results and classical work by Mahler [5] on the rational approximations to π. Let F denote the class of continuous, decreasing functions f : [, ) R such that lim f(x) = 0, x f(x) dx =. When f F, the alternating series n ( )n f(n) is conditionally convergent. Our goal is to describe the convergence set of the related series ( ) n f(n) sin(nπα), (2) α is a real number. It is natural to start one s investigation of (2) with the case when α is rational, since in that case the sequence {( ) n sin(nπα) } is periodic, and one may hope to see a pattern. Indeed, this turns out to be the case, and one quickly discovers the following result. Theorem. Suppose that f F and that α = p/q, with p Z, q Z +, and gcd(p, q) =. Then the series (2) converges if and only if q is odd.

2 Given Theorem, one may expect that when α is irrational, the convergence of (2) should depend on the nature and quality of the rational approximations to α. The next result sheds some light on the dependence. Theorem 2. Suppose that f F and α Q, and let {q n } be the sequence of denominators of convergents to the continued fraction of α. If the series qn+ f(x) dx (3) converges, then so does the series (2). q n even q 2 n By combining Theorem 2 with various facts about Diophantine approximation, we obtain the following corollaries. Corollary 3. There is a set D R, with Lebesgue measure zero, such that the series (2) converges for all real α / D and all f F. Corollary 4. Suppose that f F and α is an algebraic irrationality. Then the series (2) converges. Corollary 5. The series () converges. Corollary 3 follows from Theorem 2 and Khinchin s theorem [2] on metric Diophantine approximation. In particular, the set D can be chosen to be the set of real α such that the inequality q n+ qn 3/2 fails for infinitely many n. Similarly, Corollary 4 follows from Theorem 2 and Roth s celebrated result [8] on Diophantine approximations to algebraic irrationalities. Finally, to derive Corollary 5 directly from Theorem 2, we need to establish the convergence of (3) in the special case when α = /π and f(x) = x. To that end, it suffices to know that the sequence {q n } of denominators of convergents to the continued fraction of π satisfies an inequality of the form q n+ qn C for some absolute constant C. The first such results were obtained by Mahler in the early 950s. In particular, he showed in [5] that C = 4 is acceptable. More recent improvements on Mahler s work by Hata [] and Salikhov [9] allow us to take C even smaller (C = 7.02 and C = 6.6, respectively), but that has no effect on Corollary 5. The interested reader can easily obtain further corollaries along the lines of Corollaries 4 and 5 by appealing to other results in transcendental number theory. For example, one can use irrationality measures of numbers of the form ln a or (ln a)/(ln b), with a and b real algebraic, to obtain other explicit examples of convergent series similar to (). Theorem 2 provides a sufficient condition for convergence of alternating series of the form (2). It is natural to ask how far is this condition from being also necessary. A closer look at the special case f(x) = x θ, 0 < θ, reveals that sometimes the convergence of (3) is, in fact, equivalent to the convergence of (2). We have the following result. Theorem 6. Suppose that α Q, and let {q n } be the sequence of denominators of convergents to the continued fraction of α. When 0 < θ, the series ( ) n sin(nπα) n θ (4) converges if and only if the series does. q n even q 2 n qn+ 2 x θ dx (5)

3 In particular, it follows from Theorem 6 that the divergence set of (2) can be uncountable. Let a, b be integers, with gcd(a, b) = and b 0, and {d k } k= be an infinite sequence of s and 3 s, and consider the series λ = a b + d k 0 k!. k= When a = 0 and d k = for all k, this is the Liouville constant, the first example of a transcendental number constructed by Liouville [4]. It is not difficult to show that when λ is of the above form, the respective series (5) diverges for 0 < θ <. Furthermore, this classical example can be easily modified to construct numbers for which the series (5) converges even when θ =. Since the set L of such λ s is uncountable and dense in R, we obtain the following corollary. Corollary 7. There is an uncountable set L R, dense in R, such that the series (4) diverges for all α L and all θ (0, ]. As is stated in the opening paragraph, the present work is motivated by a calculus question. Thus, the author decided to keep the mathematical prerequisites of the paper relatively low. The statements of the theorems should be accessible to a good undergraduate student familiar with calculus and elementary number theory, and the statements of the corollaries require only slightly higher mathematical sophistication. Some high-powered mathematics is involved in the proofs of the results on Diophantine approximation used to derive the corollaries from the main theorems, but the proofs of the theorems themselves do not require advanced techniques. It is, perhaps, possible to go further by using more sophisticated harmonic analysis, but the author deliberately avoided going down that road. For example, one may be tempted to generalize the proof of Theorem 6 as to turn Theorem 2 into a necessary and sufficient condition similar to Theorem 6. However, such an attempt leads to an analysis of the asymptotic behavior as ξ 0 of the Fourier transform ˆf(ξ) of a function f F. While the author is quite interested to see what the proper such version of Theorem 2 looks like, he decided that this paper is not the place for it. Acknowledgment. The author would like to thank Geoffrey Goodson, Alexei Kolesnikov, Pencho Petrushev, and Houshang Sohrab for several conversations, suggestions and comments during the writing of this paper. 2. Preliminaries We start by introducing some notation. Throughout the remainder of the paper, we write e(z) = e 2πiz and x for the distance from the real number x to the nearest integer. We also use Landau s big-o notation and Vinogradov s -notation: if B > 0, we write A = O(B) or A B, if there exists a constant c > 0 such that A cb. We remark that in our proofs, the real numbers α, θ and the function f F are considered fixed, and so implied constants are allowed to depend on those; dependence on other parameters will be stated explicitly when it occurs. We need to recall some basic facts about continued fractions. For definitions, proofs and further details, see Khinchin s book [3]. Let α be an irrational real number. Then α has a unique representation as an (infinite) continued fraction α = [a 0 ; a, a 2,... ] = a a +. a 2 +

4 Let p n /q n, gcd(p n, q n ) =, denote the nth convergent to the continued fraction of α, p n = a 0 + q n a + + a n Then, by a variant of Khinchin [3, Theorem ], the sequences {p n } and {q n} can be defined recursively by the conditions p 0 =, q 0 = 0, p = a 0, q =, p n+ = a n p n + p n, q n+ = a n q n + q n (n ). It then follows (see Khinchin [3, Theorems 9 and 3]) that < 2q n q n+ α p n <. (6) q n q n+ Furthermore, the denominators q n grow exponentially and satisfy q n. q n 2 n/2 (n ). (7) ext, we state some estimates that will be needed in the sequel. Our first lemma is well-known to number theorists; see Montgomery [6, pp ], for example. Lemma. Suppose that is a positive integer and α R \ Z. Then ( ) e(αn) min,. (8) 2 α n=0 Lemma 2. Suppose that ν < µ and θ > 0. Then Proof. Partial integration gives and (9) follows. µ ν µ ν t θ cos t dt ν θ. (9) t θ cos t dt = t θ sin t µ ν + θ µ Lemma 3. Suppose that 0 < θ < and 0 < ν < < µ. Then A θ = µ ν 0 ν sin t dt, tθ+ t θ cos t dt = A θ + O ( µ θ + ν θ), (0) t θ cos t dt = Γ( θ) sin(πθ/2) > Proof. Inequality (0) follows from Lemma 2 and the bound ν t θ ν cos t dt t θ dt ν θ. 0 0 θ θ. () The closed-form expression for A θ is a standard Fourier cosine-transform formula. Its most natural proof (which uses the theory of contour integration) can be found in the solution of Problem III.5 in Pólya and Szegö [7, p. 33]. To justify the final inequality in (), we use that sin(πθ/2) > θ (by the concavity of the sine function) and ( θ)γ( θ) = Γ(2 θ) Γ() =. 4

5 3. Proof of Theorem We derive the theorem from Cauchy s criterion. Consider the sum S(α; M, ) = n=+ ( ) n f(n) sin(nπα), (2) M, are positive integers. Splitting S(p/q; M, ) according to the residue class of n modulo q, we have q S(p/q; M, ) = ( ) n f(n) sin(πpn/q) = h= n=+ n h (mod q) q sin(πph/q) h= We now consider separately the cases of odd and even q. n=+ n h (mod q) ( ) n f(n). (3) Case : q odd. Then the sum over n in (3) is alternating, and hence, bounded by its first term, which is at most f(). Thus, S(p/q; M, ) qf(). This establishes the convergence case of the theorem. Case 2: q even. Then p is odd and we have ( ) n = ( ) h = ( ) ph in the sum over n in (3). Hence, q S(p/q; M, ) = ( ) ph sin(πph/q) f(n). h= n=+ n h (mod q) We apply Euler s summation formula to the sum over n and deduce that S(p/q; M, ) = I q f ( ) ph sin(πph/q) + O (qf()), (4) q h= I f = I f (M, ) = f(x) dx. Since gcd(p, q) =, the sets {p, 2p,..., qp} and {, 2,..., q} are equal modulo q. Thus, q q q ( ) ph sin(πph/q) = ( ) h sin(πh/q) = sin(πh( + /q)). h= We evaluate the last sum using the well-known formula q sin((q + )x/2) sin(qx/2) sin(hx) =, sin(x/2) h= h= and (after some simplification) we obtain q ( ) ph sin(πph/q) = tan (π/2q). (5) h= 5 h=

6 Substituting (5) into the right side of (4), we conclude that S(p/q; M, ) = C q I f (M, ) + O (qf()), C q = q tan(π/2q) < 0. Since f(x) dx diverges, this completes the proof of the divergence case. 4. Proof of Theorem 2 To prove Theorem 2, we again estimate the sum S(α; M, ) defined by (2). It is convenient to assume that is even as we may, since S(α; M, ) = S(α; M, + ) + O (f()). We start by expanding the function sin(nπα) in a Fourier series. It is known and easily verified that the Fourier expansion sin(x/2) = 2 π 4 cos(kx) π 4k 2 is valid for x π, and consequently by periodicity for all real x. It follows that S(α; M, ) = 2 π The contribution from k = 0 is 2 π k= k= n=+ 4k 2 k= n=+ ( ) n f(n) f(). n=+ ( ) n f(n)e(αkn). Thus, combining the terms with k = ±m, m, we obtain S(α; M, ) = 4 { π Re } 4k 2 ( ) n f(n)e(αkn) + O (f()). (6) We estimate trivially the contribution to the right side of (6) from terms with k > M, and we get S(α; M, ) = 4 { M π Re e(αk) M } 4k 2 ( ) n g(n)e(αkn) + O (f()), (7) k= g(x) = f( + x). By partial summation, M ( ) n g(n)e(βn) = g(m)u(β; M) g(m) = g(m + ) g(m) and m U(β; m) = ( ) n e(βn) = M m= g(m)u(β; m), (8) m e((β + /2)n). Substituting (8) into the right side of (7), we obtain S(α; M, ) = 4 M {g(m)v π Re } (α; M) g(m)v (α; m) + O(f()), (9) 6 m=

7 V (α; m) = M k= e(αk) 4k 2 U(kα; m). In order to estimate the right side of (9), we break the sum V (α; m) into blocks depending on the denominators of the rational approximations to α. Let {p n /q n } n=0 be the sequence of convergents to the continued fraction of α. We decompose V (α; m) into blocks V j (α; m) defined by V j (α; m) = k K j (M) e(αk) 4k 2 U(kα; m), (20) K j (M) is the set of positive integers k M such that q j < 4k q j+. We obtain different estimates for V j (α; m), depending on the size and parity of q j. 4.. Estimation of V j (α; m) for small j. When j is bounded above by an absolute constant, we appeal to (8) and get kα + V j (α; m) /2 8k 2 = K j (α), say. (2) 2 q j <4k q j Estimation of V j (α; m) for odd q j. Suppose that q j is odd and sufficiently large, and write p = p j, q = q j, r = q j+, and β = α p/q. From (6), when 4k r, k β < k qr 4q. Since q is odd, we have 2q (2pk + q) and δ p,q (k) = 2pk + q 2q 2q. Hence, Using (2), we obtain kα + 2 δ p,q(k) k β δ p,q (k) 4q δ p,q(k). 2 V j (α; m) q<4k r = h 2q 2 h δ p,q (k) 4k 2 = h 2q h 2q 2 h q<4k r 2pk h (mod q) q<4k r 2pk+q h (mod 2q) δ p,q (k) 4k 2 4k 2. (22) Upon noting that the sum over k on the right side of (22) is O(q 2 ), we deduce from (22) that q V j (α; m) q 2 2h 2q h= (q+)/2 q h= 7 2h q j ln q j. (23)

8 4.3. Estimation of V j (α; m) for even q j. Suppose that q j is even, and let p, q, r, and β be as in 4.2. Except when 2k q (mod 2q), we can argue similarly to 4.2. Hence, when q is even, we get When 2k q (mod 2q), we get so (8) gives Therefore, by (25), V j (α; m) = V j (α; m) + O ( q ln q ), (24) V j (α; m) = k K j (M) 2k q (mod 2q) U(kα; m) = e(αk) 4k 2 m e(βkn), U(kα; m) min ( m, (2k β ) ) min (m, r). V j (α; m) q<2l r l q (mod 2q) Combining (24) and (26), we conclude that min(m, r) l 2 U(kα; m). (25) q 2 min(m, r). (26) V j (α; m) qj 2 min(m, q j+ ) + qj ln q j. (27) Moreover, we note that the first term on the right side of (27) is superfluous when 2q j > q j+, since in that case the sum V j (α; m) is empty Completion of the proof. Let j 0 2 be an integer to be chosen later, and set K = j 0 j=0 K j (α) = kα + /2 8k k q j0 We use (2) to estimate the contribution to V (α; m) from subsums V j (α; m) with j < j 0, and we use (23) and (27) to estimate the contribution from sums V j (α; m) with j j 0. Let I α (M) denote the set of indices j j 0 such that q j is even and satisfies the inequalities q j M and 2q j q j+. We obtain ln q j V (α; m) K + c + c qj 2 min(m, q j+ ), q j j j 0 j I α(m) c > 0 is an absolute constant. Recalling (7), we deduce V (α; m) K + c 2 + 2c qj 2 min(m, r j ), (28) j I α(m) c 2 > 0 is another absolute constant and r j = q j+ /2. Using (28) to bound the right side of (9), we get S(α; M, ) 3c qj 2 Σ j + O j0 (f()), (29) j I α(m) the O-implied constant depends on j 0 and M Σ j = g(m) min(m, r j ) g(m) min(m, r j ). m= 8

9 Using partial summation and the monotonicity of f, we find that Σ j rj qj+ g(n) g(x) dx f(x) dx. n r 0 j Hence, (29) yields S(α; M, ) j=j 0 q j even j I α(m) q 2 j 3c q 2 j qj+ f(x) dx + O j0 (f()). (30) Finally, let us fix an ε > 0. Since the series (3) converges, we can find an index j 0 = j 0 (ε) such that qj+ f(x) dx < ε. 6c We choose j 0 above to be such an integer and fix it. Then qj+ f(x) dx and (30) yields q 2 j I α(m) j j=j 0 q j even q 2 j qj+ S(α; M, ) < ε 2 + O ε(f()), f(x) dx < the O-implied constant depends on ε. Therefore, we can find an integer 0 = 0 (ε, α, f) such that when 0, one has S(α; M, ) < ε. This establishes the convergence of the series (2). 5. Proof of Theorem 6 We assume that the series (5) diverges and consider the sum S(α; M, ) one last time. We will use (9) to show that S(α; M, ) can approach as M,. We retain the notation introduced in the proof of Theorem 2 and proceed with the estimation of S(α; M, ). Let j 0 2 be a fixed integer chosen so that q j0 is sufficiently large, and let be a large even integer. We restrict the choice of to integers of the form q j b, with j > j 0 and b {, 2}. Using (9), (2), (23), (24), and (27), we obtain the following version of (29): S(α; M, ) = S j (α; M, ) + O ( θ), (3) j I α(m) ε 6c, I α (M) is the set of indices defined in 4.4, S j (α; M, ) = 4 { M π Re } g(m)v j (α; M) g(m)v j (α; m), and V j (α; m) is the sum defined by (25). Furthermore, by (26) and the choice of, for indices j with q j, we have V j (α; M, ) qj 2 q j+ qj 2, whence S j (α; M, ) qj 2. Thus, from (3), S(α; M, ) = S j (α; M, ) + O(), (32) j I α(m,) 9 m=

10 I α(m, ) is the set of indices j I α (M) such that q j >. We now proceed to obtain an approximation for V j (α; m), which we will then use to estimate the right side of (32). Let p, q, r and β be as in 4.3. When 2k q (mod 2q), we have m U(kα; m) = e(βkn) = e(kmβ) e(kβ) + O(). Using the Taylor expansion e(z) = + 2πiz + O( z 2 ), we deduce that when k r, U(kα; m) = e(kmβ) 2πikβ We substitute this approximation in (25) and obtain V j (α; m) = k L j (M) = k L j (M) e(k β/2) k 2 k 2 +m + O(). e(kmβ/2) πikβ + O ( q 2) e(ktβ/2) dt + O ( q 2), L j (M) denotes the set of even integers k such that 2 k K j(m) and k q (mod 2q). Hence, S j (α; M, ) = 4 Re Ξ k (α; M, ) π k 2, k L j (M) M Ξ k (α; M, ) = g(m) e(ktβ/2) dt g(m) m= +m Recall that here g(x) = ( + x) θ. Using partial summation, we find that Hence, S j (α; M, ) = 4 π Ξ k (α; M, ) = = 4 π = k L j (M) k L j (M) g( t )e(ktβ/2) dt t θ e(ktβ/2) dt + O ( θ). e(ktβ/2) dt. k 2 t θ cos(πktβ) dt + O ( q 2 θ) (πk β ) θ k 2 νk +µ k ν k t θ cos t dt + O ( q 2 θ), (33) ν k = πk β and µ k = πk β M. Summing over j, we deduce from (32) and (33) that S(α; M, ) = 4 (πk β j ) θ νk +µ k π k 2 t θ cos t dt + O (), (34) ν k j I α(m,) k L j (M) β j = qj q j α. In order to estimate the right side of (34), we will impose some restrictions on the choice of M. Let Q α be the set of even members of {q n } such that qn+ x θ dx > q θ/2 n. 0

11 We remark that the contribution to the series (5) from terms with q n / Q α is dominated by the convergent series q q θ/2. Thus, the divergence of (5) implies the divergence of the series qn+ q 2 x θ dx. (35) q n n Q α In particular, the set Q α is infinite. We restrict M to the sequence of numbers of the form q +θ/3, with q Q α. Let J = J(M, ) denote the largest index in the set I α(m, ). Using our restriction on the choice of M and the bound νk +µ k ν k t θ cos t dt we find that the term with j = J in (34) is q 2k M/2 k q (mod 2q) { µ θ k if θ <, ln M if θ =, q θ/2 k 2 q, q = q J and the implied constant depends on θ. Hence, S(α; M, ) = 4 (πk β j ) θ νk +µ k π k 2 t θ cos t dt + O (), (36) ν k j I α(m,) k L j (M) I α(m, ) = I α(m, ) \ {J}. Since the integrals on the right side of (36) behave somewhat differently when θ = and when 0 < θ <, we now consider these two cases separately. 5.. The case θ =. When j I α(m, ) and k L j (M), we have µ k = πk β j M >. Hence, by Lemma 2, νk +µ k ν k t cos t dt = From this inequality and (36), we obtain S(α; M, ) = 4 π 4 π ν k t cos t dt + O() = t ( + O ( t 2)) dt + O() ν k = ln ν k + O() = ln q j+ + O(ln(k)). j I α(m,) k L j (M) j I α (M,) q n Q α (M,) q 2 n ln q j+ k 2 + O () ln q j+ qj 2 + O() qn+ dt t + O(), (37) Q α(m, ) is the set of those q n Q α for which q n > and q n+ M. In view of the divergence of the series (35), this establishes that lim sup S(α; M, ) =. M

12 5.2. The case 0 < θ <. When j I α(m, ) and k L j (M), by Lemma 3, νk +µ k ν k t θ cos t dt = A θ + O ( ν θ k + µ θ k A θ is the Fourier integral (). From this inequality and (36), we obtain S(α; M, ) = 4A θ (πk β j ) θ π k 2 + O ( + ) By (6) and the restriction on M, M θ 4A θ π 2 θ = M θ j I α(m,) k L j (M) j I α(m,) j I α(m,) q θ j+ q 2 j j I α (M,) k L j (M) + O ( + ), q j+ q 2 j M γ (k β j ) k 2. j I α(m,) γ = 4 θ2 > 0. Thus, for sufficiently large values of M, we obtain S(α; M, ) 2A θ π 2 θ A θ j I α(m,) q θ j+ q 2 j + O (), q q n Q n 2 α(m,) qn+ ), q θ j+ q 2 j, t θ dt + O (), (38) A θ = 2πθ 2 ( θ)a θ > 0 and Q α(m, ) is defined as in 5.. Therefore, once again, using (38) and the divergence of the series (35), we conclude that lim sup S(α; M, ) =. M This completes the proof of the theorem. References [] M. Hata, Rational approximations to π and some other numbers, Acta Arith. 63 (993), [2] A.Ya. Khinchin, Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen, Math. Ann. 92 (924), [3] A.Ya. Khinchin, Continued Fractions, Dover Publications, 997, translated from the 3rd Russian edition. [4] J. Liouville, Sur des classes très-étendues de quantités dont la valeur n est ni algébrique, ni même réductible à des irrationelles algébriques, J. Math. Pures Appl. 6 (85), [5] K. Mahler, On the approximation of π, Indag. Math. 5 (953), [6] H.L. Montgomery, Ten Lectures on the Interface Between Analytic umber Theory and Harmonic Analysis, American Mathematical Society, Providence, RI, 994. [7] G. Pólya and G. Szegö, Problems and Theorems in Analysis, vol. I, Springer Verlag, 972. [8] K.F. Roth, Rational approximations to algebraic numbers, Mathematika 2 (955), 20. [9] V.Kh. Salikhov, On the irrationality measure of π, Russian Math. Surveys 63 (2008), , translated from Russian. Department of Mathematics, 7800 York Road, Towson University, Towson, MD address: akumchev@towson.edu 2