27 citations · 81 across the 27 of their papers we have counts for
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A Quadratic Harmonic Approximation
Mark B. Villarino
We obtain the quadratic term in Euler's asymptotic expansion of the nth harmonic number by a simple modification of Young's elementary determination of the linear term.
The error in an alternating series
Mark B. Villarino
We give a new proof of Johnsonbaugh's refined error estimates of an alternating series based on an idea of R. M. Young. We also give a new proof of the error estimate and convergen…
On the Convergence of the Catalan Generating Function
Mark B. Villarino
We solve a problem of Marshal Hall by proving the convergence of the catalan number generating function directly on the basis of its recursion formula.
Quotient polynomials with positive coefficients
Mark B. Villarino
We give an optimal necessary and sufficient condition for the quotient polynomial and remainder in the division algorithm to have positive coefficients.
Algebraic Addition Theorems
Mark B. Villarino
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…
A Refinement of Ramanujan's Factorial Approximation
Michael D. Hirschhorn, Mark B. Villarino
We give an elementary proof for new strict upper and lower bounds for the correction term in Ramanujan's approximation for the factorial function
An integral inequality and the Ricatti-Bernoulli differential equation
Mark B. Villarino
We apply an integral inequality to obtain a rigorous \emph{a priori} estimate of the accuracy of the partial sum to the power series solution of the Ricatti-Bernoulli differential…
On the accuracy of the Chakrabarti-Hudson approximation to
Mark B. Villarino
We obtain rigorous upper and lower bounds for the error in the recent approximation for proposed by Chakrabarti and Hudson
The AGM Simple Pendulum
Mark B. Villarino
We present a self-contained development of Gauss' Arithmetic-Geometric Mean (AGM) and the work of A.E. Ingham who obtained rigorous error bounds for the AGM approximations to the p…
On the monotonicity of the correction term in Ramanujan's factorial approximation
Mark B. Villarino, Daniel Campos Salas, Javier Carvajal Rojas
We present two new proofs of the monotonicity of the correction term in Ramanujan's refinement of Stirling's formula.
The Error in Rayleigh's Approximative Period
Mark B. Villarino
We obtain rigorous a priori upper and lower bounds to the exact period of the celebrated Rayleigh stretched string differential equation. We use them to show that Rayleigh's approx…
An Integral Inequality and the Riccati-Bernoulli Differential Equation
Mark B. Villarino
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 differenti…
Ramanujan's Harmonic Number Expansion into Negative Powers of a Triangular Number
Mark B. Villarino
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 t…
Ramanujan's Harmonic Number Expansion
Mark B. Villarino
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 t…
Sharp Bounds for the Harmonic Numbers
Mark B. Villarino
We obtain best possible upper and lower bounds for the Lodge-Ramanujan and DeTemple-Wang approximations to the nth Harmonic Number
A Direct Proof of Landen's Transformation
Mark B. Villarino
We prove Landen's transformation by a direct elementary transformation of infinite series
Ramanujan's Inverse Elliptic Arc Approximation
Mark B. Villarino
We suggest a continued fraction origin to Ramanujan's approximation to {(a-b)/(a+b)}^2 in terms of the arc length of an ellipse with semiaxes a and b. Moreover, we discuss the asym…
Ramanujan's Perimeter of an Ellipse
Mark B. Villarino
We present a detailed error analysis of Ramanujan's most accurate approximation to the perimeter of an ellipse.
Ramanujan's Approximation to the nth Partial Sum of the Harmonic Series
Mark B. Villarino
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…