Some New Facts in Discrete Asymptotic Analysis

Transcription

1 Mathematica Balkanica New Series Vol 2, 2007, Fasc 3-4 Some New Facts in Discrete Asymptotic Analysis Cristinel Mortici, Andrei Vernescu Presented by P Boyvalenkov This paper is closely related to [7] Here, the convergence of a logarithmic sum is presented which generalizes a result from a previous paper [7] of this Journal An interesting estimation 22 for the quantity ln lnn + ln ln n is established and then a more general result is obtained Then, an inequality concerning the constant of Euler is refined A simpler proof for the rate of convergence of a class of sequences studied in [7] is also given Finally, an estimation for the sequence converging to the Euler s constant γ is given, which is stronger than the estimation established in [] by J Franel Introduction In a previous paper [7], appeared in Mathematica Balkanica, devoted to the discrete asymptotic analysis, one of the authors has presented the asymptotic description of first order of convergence of the sequence S n n 2 with general term S n = log n n + This is 2 S n =n + ln ln n + A + o, where A is a certain constant ψ This constant is A = α + β =0, ,where nx α = lim S n n = 0, n k ln k and! β = lim k ln k ln ln n =0, n ψ nx!

2 302 C Mortici, A Vernescu Now, we establish the convergence of two sequences which generalize, namely 3 u n = and 4 v n = + a k ln ln + r a k ln ln where n is an arithmetic progression with a andratior 0, ] the sequence 3 is the special case for r = 2 In order to establish the announced results, we begin by presenting two convenient inequalities [22 and 3] We use the well-known inequality of Neper 2 x + < ln + < x x, for all positive real numbers x, or, denoting + x = t, t t < ln t<t, for all real numbers t, So, we have the following Lemma 2 For all positive integers n 2, the following inequality 22 log n+ + < ln lnn + ln ln n<log n + n + n holds true P r o o f For the first inequality we succesively have ln lnn + ln ln n =ln lnn + ln n < lnn + ln n = More generally, the inequality is valid for al real numbers x, 0,, but we use it only for x0

3 Some New Facts in Discrete Asymptotic Analysis 303 = lnn + ln n ln n = ln + n ln n =log n + n Similarly, to prove the second part, we observe that by the theorem of Lagrange, we can find c n n, n +forwhich ln lnn + ln ln n = c n ln c n Further, c n ln c n n +lnn + = n + lnn + ln + n + lnn + =log n+ + n + The inequality is proved 3 Let us consider an arithmetic progression n with a andthe ratio r 0, ] Then we have Lemma 3 For all integers n 2, the following inequality 3 log an+ + < ln ln + ln ln < log a an + r n+ holds true Proof First, we have ln ln + ln ln =ln ln + ln < ln + ln = = ln + ln ln = ln an+ ln =log an + =log an + r Then, by Lagrange s theorem, we can find c n,+ sothat ln ln + ln ln = c n ln c n

4 304 C Mortici, A Vernescu Further, c n ln c n ln + + = + ln + ln = ln ln + ln + =log an+ + + Corollary 32 For 0 <r<, we have 32 log an+ + < ln ln + ln ln < log a an + n+ and 33 log an+ + r < ln ln + ln ln < log a an + r n+ Theorem 33 Let there be given an arithmetic progression n with a andratior 0, ] Then, the sequence u n n given by u n = is decreasing and bounded P r o o f First, using 32, we have + a k ln ln n+ u n+ u n = + ln ln + a k + a k +lnln = =log an+ + ln ln + ln ln < 0 + Now, using the left-hand side of the inequality 32, we prove that the sequence u n n is bounded from below, ie u n = + a k ln ln ln ln a k+ ln ln a k ln ln = ln ln + ln ln a 2 ln ln =

5 Some New Facts in Discrete Asymptotic Analysis 305 = ln ln + ln ln ln ln a 2 ln ln a 2 According to the Weierstrass theorem, the sequence u n n is convergent Theorem 34 Let there be given an arithmetic progression n with a andratior 0, ] Then, the sequence v n n given by v n = is decreasing and bounded P r o o f First, using 33, we have + r a k ln ln n+ v n+ v n = + r ln ln + a k + r a k +lnln = =log an+ + r ln ln + ln ln < 0 + Now, using the left-hand side of the inequality 33, we prove that the sequence v n n is bounded from below, v n = + r a k ln ln ln ln a k+ ln ln a k ln ln = ln ln + ln ln a 2 ln ln = = ln ln + ln ln ln ln a 2 ln ln a 2 According to the Weierstrass theorem, the sequence v n n is convergent Corollary 35 The sequence w n n given by the formula w n = is decreasing and bounded log n + ln ln n n

6 306 C Mortici, A Vernescu 4 Rate of Convergence We add now a little new fact concerning the rate of convergence of the sequence γ n n, given by the formula γ n = ln n, n to its limit γ Euler s constant In [5] the inequality 4 2n + <γ n γ< 2n is proved Later new sequences faster convergent to γ were introduced, for example the sequence H n n given by the formula R n = H n ln n +, 2 where H n n denotes the harmonic sum sequence H n = n The sequence R n n converges decreasingly to γ with the speed 42 24n + 2 <R n γ< 24n 2 For proofs and other comments, see [3] In [4], the sequence T n n given by the formula T n = H n ln n n is also defined which converges increasingly to γ Moreover, 43 48n + 3 <γ T n < 48n 3 In [5], the sequence x n n given by the formula x n = n + ln n 2n

7 Some New Facts in Discrete Asymptotic Analysis 307 is also considered Note that the author had the idea to defined the sequence x n n by replacing the term /n by /2n in the sequence γ n n, x n = γ n n + 2n = γ n 2n The sequel is that the new sequence x n n converges faster to γ Indeed, the following estimations 44 2n + 2 <γ n x n < 2n 2 hold for all integers n 2, so the order convergence of the sequence x n n is /2n 2 This fact is closely related to the asymptotic development of the harmonic sum H n, 45 H n =lnn + γ + 2n 2n n 4 ε n, with 0 <ε n < 252n 2 In the same way we can see that every other replacement of the term /n by α/n, with α /2, leads to a weaker convergence, because the term /2n from 45 dissapears only in the case α =/2, H n ln n 2n = 2n n 4 ε n The estimations 44 allow us to establish the better estimations 46 2n 2n 2 <γ n γ< 2n 2n + 2 for the sequence γ n n In this way, from the identity we deduce Hence, x n = γ n 2n 2n + 2 <γ γ n < 2n 2n 2 2n + 2 2n <γ γ n < 2n 2 2n,

8 308 C Mortici, A Vernescu so that 46 is proved Now, mention that the estimations 46 we obtained here are stronger than the estimations 2n 8n 2 <γ n γ< 2n due to J Franel eg [], p 523, because and, obviously, 2n 2n 2 2n 8n 2 2n 2n + 2 < 2n References [] K K n a p p, Theory and applications in infinite series, 2nd edition, London- Glasgow, Blackie & Son, 964 [2] C M o r t i c i, A V e r n e s c u, An improvement of the convergence speed of the sequence γ n n converging to Euler s constant, Analele Univ Ovidius Constanta, accepted [3] D W d e T e m p l e, A quicker convergence to Euler s constant, Amer Math Monthly, , [4] T N e g o i, A more fast convergence to Euler s constant, Gazeta Matematica A, 5 997, no 2, -3 in Romanian [5] A V e r n e s c u, The order of convergence of the sequence which define the Euler s constant, Gazeta Matematica, 983, in Romanian [6] A V e r n e s c u, A new accelerate convergence to the constant of Euler, Gazeta Matematica A, 7 999, [7] A V e r n e s c u, Some Aspects in Discrete Asymptotic Analysis, Mathematica Balkanica, New Series, , Fasc -2, Valahia University of Targoviste Received Department of Mathematics Bd Unirii 8, Targoviste, Romania cmortici@valahiaro, avernescu@valahiaro