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Ramanujan and Euler's Constant
"... We consider Ramanujan's contribution to formulas for Euler's constant fl. For example, in his second notebook Ramanujan states that (in modern notation) 1 X k=1 (\Gamma1) k\Gamma1 nk ` x k k! ' n = ln x + fl + o(1) as x ! 1. This is known to be correct for the case n = 1, but incorrect f ..."
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We consider Ramanujan's contribution to formulas for Euler's constant fl. For example, in his second notebook Ramanujan states that (in modern notation) 1 X k=1 (\Gamma1) k\Gamma1 nk ` x k k! ' n = ln x + fl + o(1) as x ! 1. This is known to be correct for the case n = 1, but incorrect for n ? 2. We consider the case n = 2. We also suggest a different, correct generalization of the case n = 1. 1.
S.: Passages of proof
 Bull. Eur. Assoc. Theor. Comput. Sci. EATCS
, 2004
"... Whether ’tis nobler in the mind to suffer The slings and arrows of outrageous fortune, Or to take arms against a sea of troubles And by opposing end them? Hamlet 3/1, by W. Shakespeare In this paper we propose a new perspective on the evolution and history of the idea of mathematical proof. Proofs w ..."
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Whether ’tis nobler in the mind to suffer The slings and arrows of outrageous fortune, Or to take arms against a sea of troubles And by opposing end them? Hamlet 3/1, by W. Shakespeare In this paper we propose a new perspective on the evolution and history of the idea of mathematical proof. Proofs will be studied at three levels: syntactical, semantical and pragmatical. Computerassisted proofs will be give a special attention. Finally, in a highly speculative part, we will anticipate the evolution of proofs under the assumption that the quantum computer will materialize. We will argue that there is little ‘intrinsic ’ difference between traditional and ‘unconventional ’ types of proofs. 2 Mathematical Proofs: An Evolution in Eight Stages Theory is to practice as rigour is to vigour. D. E. Knuth Reason and experiment are two ways to acquire knowledge. For a long time mathematical
A Survey of Euler’s Constant
, 2009
"... doi:10.4169/193009809X468689 (∑n The mathematical constant γ = limn→∞ k=1 1 k − ln(n)) = 0.5772156..., known as Euler’s constant, is not as well known as its cousins π, e, i, but is still important enough to warrant serious consideration in the circles of applied mathematics, calculus, and number t ..."
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doi:10.4169/193009809X468689 (∑n The mathematical constant γ = limn→∞ k=1 1 k − ln(n)) = 0.5772156..., known as Euler’s constant, is not as well known as its cousins π, e, i, but is still important enough to warrant serious consideration in the circles of applied mathematics, calculus, and number theory. Some authors will occasionally refer to γ as the EulerMascheroni constant, so named after the Italian geometer Lorenzo Mascheroni (1750–1800), who actually introduced the symbol γ for the constant (although there is controversy about this claim) and also computed, though with error, the first 32 digits [16, 34]. Sometimes one will find in older texts the symbols C (this was Euler’s constant of integration) and A (also from Mascheroni) to represent the constant, but these notations seem to have disappeared in the modern era [27]. Our aim in this article is to present a survey of γ that is both manageable by, and enlightening to, those who favor mathematics at the undergraduate level. To try and follow in the footsteps of the big boys π and e is quite a chore, but this brief historical description of γ and colorful portfolio of applications and surprising appearances in a multitude of settings is both impressive and mathematically educational. Defining and evaluating the constant Calculus students can approximate the integral ∫ n (1/x) dx = ln(n) by inscribed and 1 circumscribed rectangles, and hence obtain the inequalities (for any integer n> 1) 1 n < n ∑ 1 − ln(n) <1, k so if the limit exists, 0 ≤ lim n→∞ k=1 n∑ k=1
EULER’S CONSTANT: EULER’S WORK AND MODERN DEVELOPMENTS
, 2013
"... Abstract. This paper has two parts. The first part surveys Euler’s work on the constant γ =0.57721 ·· · bearing his name, together with some of his related work on the gamma function, values of the zeta function, and divergent series. The second part describes various mathematical developments invol ..."
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Abstract. This paper has two parts. The first part surveys Euler’s work on the constant γ =0.57721 ·· · bearing his name, together with some of his related work on the gamma function, values of the zeta function, and divergent series. The second part describes various mathematical developments involving Euler’s constant, as well as another constant, the Euler–Gompertz constant. These developments include connections with arithmetic functions and the Riemann hypothesis, and with sieve methods, random permutations, and random matrix products. It also includes recent results on Diophantine approximation and transcendence related to Euler’s constant. Contents
Rearranging the Alternating Harmonic Series
"... Why are conditionally convergent series interesting? While mathematicians might undoubtably give many answers to such a question, Riemann’s theorem on rearrangements of conditionally convergent series would probably rank near the top of most responses. Conditionally convergent series are those serie ..."
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Why are conditionally convergent series interesting? While mathematicians might undoubtably give many answers to such a question, Riemann’s theorem on rearrangements of conditionally convergent series would probably rank near the top of most responses. Conditionally convergent series are those series that converge as written, but do not converge when each of their terms is replaced by the corresponding absolute value. The nineteenthcentury mathematician Georg Friedrich Bernhard Riemann (18261866) proved that such series could be rearranged to converge to any prescribed sum. Almost every calculus text contains a chapter on infinite series that distinguishes between absolutely and conditionally convergent series. Students see the usefulness of studying absolutely convergent series since most convergence tests are for positive series, but to them conditionally convergent series seem to exist simply to provide good test questions for the instructor. This is unfortunate since the proof of Riemann’s theorem is a model of clever simplicity that produces an exact algorithm. It is clear, however, that even with such a simple example as the alternating harmonic series one cannot hope for a closed form solution to the problem of rearranging it to sum to an arbitrary real number. Nevertheless, it is an old result that for any real number of the form ln