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Search Results: 1 - 10 of 160655 matches for " Mark B. Villarino "
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Rayleigh's Stretched String
Mark B. Villarino
Physics , 2012,
Abstract: We obtain rigorous \emph{a priori} upper and lower bounds to the exact period of the celebrated Rayleigh stretched string differential equation
Ramanujan's Approximation to the nth Partial Sum of the Harmonic Series
Mark B. Villarino
Mathematics , 2004,
Abstract: A simple integration by parts and telescopic cancellation leads to a rigorous derivation of the first 2 terms for the error in Ramanujan's asymptotic series for the nth partial sum of the harmonic series. Then Kummer's transformation gives three more terms and a rigorous error estimate. Finally best-possible estimates of Lodge's approximations.
Sharp Bounds for the Harmonic Numbers
Mark B. Villarino
Mathematics , 2005,
Abstract: We obtain best possible upper and lower bounds for the Lodge-Ramanujan and DeTemple-Wang approximations to the nth Harmonic Number
The Probability of a Run
Mark B. Villarino
Mathematics , 2005,
Abstract: We prove the explicit formula for the probability of a run of r successes in n trials.
Ramanujan's Harmonic Number Expansion
Mark B. Villarino
Mathematics , 2005,
Abstract: An algebraic transformation of the DeTemple-Wang half-integer approximation to the harmonic series produces the general formula and error estimate for the Ramanujan expansion for the nth harmonic number.
Ramanujan's Harmonic Number Expansion into Negative Powers of a Triangular Number
Mark B. Villarino
Mathematics , 2007,
Abstract: An algebraic transformation of the DeTemple-Wang half-integer approximation to the harmonic series produces the general formula and error estimate for the Ramanujan expansion for the nth harmonic number into negative powers of the nth triangular number. We also discuss the history of the Ramanujan expansion for the nth harmonic number as well as sharp estimates of its accuracy, with complete proofs, and we compare it with other approximative formulas.
An Integral Inequality and the Riccati-Bernoulli Differential Equation
Mark B. Villarino
Mathematics , 2007,
Abstract: We apply an integral inequality to obtain a rigorous apriori estimate of the accuracy of the partial sum to the power series solution of the celebrated Riccati-Bernoulli differential equation
Rayleigh's Stretched String
Mark B. Villarino
Mathematics , 2008,
Abstract: We obtain rigorous a priori upper and lower bounds to the exact period of the celebrated Rayleigh stretched string differential equation.
Algebraic Addition Theorems
Mark B. Villarino
Mathematics , 2012,
Abstract: We present a self-contained development of the Weierstrass theory of those analytic functions (single-valued or multiform) which admit an algebraic addition theorem. We review the history of the theory and present detailed proofs of the major theorems.
A Cubic Surface of Revolution
Mark B. Villarino
Mathematics , 2013,
Abstract: We develop a direct and elementary (calculus-free) exposition of the famous cubic surface of revolution x^3+y^3+z^3-3xyz=1.12 pages. We have added a second elementary proof that the surface is of revolution.
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