Related papers: A Refinement of Ramanujan's Factorial Approximatio…
We present two new proofs of the monotonicity of the correction term $\theta_n$ in Ramanujan's refinement of Stirling's formula.
In this paper we want to prove some formulas listed by S. Ramanujan in his paper "Modular equations and approximations to $\pi$" \cite{24} with an elementary method.
We give a generalization of a Ramanujan's exercise for high school students. Our results can be regarded as a variation of the factorization formula of $x^{n} - 1$.
We investigate the rational approximation of fractional powers of unbounded positive operators attainable with a specific integral representation of the operator function. We provide accurate error bounds by exploiting classical results in…
Several terminating generalizations of Ramanujan's formula for $\frac{1}{\pi}$ with complete WZ proofs are given.
We generalize Ramanujan method of approximating the smallest root of an equation which is found in Ramanujan Note books, Part-I. We provide simple analytical proof to study convergence of this method. Moreover, we study iterative approach…
In this short research note, we aim to establish an interesting extension of a summation due to Ramanujan.The result is derived with the help of an extension of Gauss's summation theorem available in the literature.
The goal of this paper is to formulate a systematical method for constructing the fastest possible continued fraction approximations of a class of functions. The main tools are the multiple-correction method, the generalized Mortici's lemma…
In this article we use theoretical and numerical methods to evaluate in a closed-exact form the parameters of Ramanujan type $1/\pi$ formulas.
Ramanujan's approximation to the exponential function is reexamined with the help of Perron's saddle-point method. This allows for a wide generalization that includes the results of Buckholtz, and where all the asymptotic expansion…
We study Ramanujan's cubic continued fraction and explicit evaluations of theta-functions
A highly strong upper estimate in the modified asymptotic formula for sums of the primes' reciprocals is proved to be necessary (as well as sufficient) in order the Ramanujan inequality holds true. Some other criteria in similar terms are…
Inspired by the recent pioneering work, dubbed "The Ramanujan Machine" by Raayoni et al. (arXiv:1907.00205), we (automatically) [rigorously] prove some of their conjectures regarding the exact values of some specific infinite continued…
In this short note we use the umbral formalism to derive the Ramanujan Master Theorem and discuss its extension to more general cases.
A conjectured relation between Ramanujan's asymptotic approximations to the exponential function and the exponential integral is established. The proof involves Stirling numbers, second-order Eulerian numbers, modifications of both of…
In this paper we present some new upper bounds of the Cusa-Huygens and the Huygens approximations. Bounds are obtained in the forms of some polynomial and some rational functions.
In this paper we present experimental ways of evaluating Ramanujan`s quantities which as someone can see are related with algebraic numbers. The good thing with algebraic numbers is that can be found in a closed form, from there…
Several continued fraction expansions for $e$ have been produced by an automated conjecture generator (ACG) called \emph{The Ramanujan Machine}. Some of these were already known, some have recently been proved and some remain unproven.…
In the paper, some lower bounds for polygamma functions are refined.
We show that the absolute convergence of a Ramanujan expansion does not guarantee the convergence of its real variable generalization, which is obtained by replacing the integer argument in the Ramanujan sums with a real number. We also…