27 citations · 81 across the 27 of their papers we have counts for
8 papers · 1 filter
The Probability of a Run
Mark B. Villarino
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
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.
On the Archimedean or Semiregular Polyhedra
Mark B. Villarino
We prove that there are thirteen Archimedean/semiregular polyhedra by using Euler's polyhedral formula.
Mertens' Proof of Mertens' Theorem
Mark B. Villarino
We present a self-contained elementary and detailed exposition of Mertens' own proof of his theorem on the divergence of the series of the reciprocals of the primes and compare it…