相关论文: On the elegance of Ramanujan's series for $\pi$

In 1914, Ramanujan presented a collection of 17 elegant and rapidly converging formulae for $\pi$. Among these, one of the most celebrated is the following series:…

In a famous paper of $1914$ Ramanujan gave a list of $17$ extraordinary formulas for the number $\pi$. In this paper we explain a general method to prove them, based on an original idea of James Wan and in some own ideas.

Via symbolic computation we deduce 97 new type series for powers of $\pi$ related to Ramanujan-type series. Here are three typical examples: $$\sum_{k=0}^\infty \frac{P(k) \binom{2k}k\binom{3k}k…

Through an application of a remarkable result due to Mishev in 2018 concerning the inverses for a class of transformations of sequences of complex numbers, we obtain a very simple proof for a famous series for $\frac{1}{\pi}$ due to…

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.

First we give general formulas for proving real or complex Ramanujan series for $1/\pi$. Then, as an example, we apply them for providing complete proofs of the fastest series for $1/\pi$ due to Ramanujan using Russell and Weber modular…

Using some properties of the gamma function and the well-known Gauss summation formula for the classical hypergeometric series, we prove a four-parameter series expansion formula, which can produce infinitely many Ramanujan type series for…

We prove a polynomial continued fraction identity for the constant $-\pi/4$, conjectured by the Ramanujan Machine project. The proof proceeds by explicitly solving the underlying second-order linear difference equation. We derive a…

A sharp asymptotic formula for the sum of reciprocals of $\pi(n)$ is derived, where $\pi(x)$ is the number of primes not exceeding $x$. This result improves the previous results of De Koninck--Ivi\'c and L. Panaitopol.

We give $q$-analogues of the following two Ramanujan-type formulas for $1/\pi$: \begin{align*} \sum_{k=0}^\infty (6k+1)\frac{(\frac{1}{2})_k^3}{k!^3 4^k} =\frac{4}{\pi} \quad\text{and}\quad \sum_{k=0}^\infty…

In this paper, we deduce a family of six new series for $1/\pi$; for example, $$\sum_{n=0}^\infty\frac{41673840n+4777111}{5780^n}W_n\left(\frac{1444}{1445}\right) =\frac{147758475}{\sqrt{95}\,\pi}$$ where $W_n(x)=\sum_{k=0}^n\binom…

About 40 years ago Jonathan and Peter Borwein discovered the series identity $$ \sum_{n=0}^\infty \frac{(-1)^n(6n)!}{(3n)!(n!)^3} \frac{(A+nB)}{C^{n+1/2}} = \frac{1}{12\pi} $$ where \begin{align*} A&=1657145277365+212175710912\sqrt{61},\cr…

In this article, we define functions analogous to Ramanujan's function $f(n)$ defined in his famous paper "Modular equations and approximations to $\pi$". We then use these new functions to study Ramanujan's series for $1/\pi$ associated…

This paper is a tribute to the genius of the legendary Indian mathematician Srinivasa Ramanujan (22 December 1887 - 26 April 1920) in the centenary year of his death. The life story of Ramanujan is so well known that it needs no elaboration…

Ramanujan proved that the inequality $\pi(x)^2 < \frac{e x}{\log x} \pi\Big(\frac{x}{e}\Big)$ holds for all sufficiently large values of $x$. Using an explicit estimate for the error in the prime number theorem, we show unconditionally that…

In this paper, we establish the irrationality of some open problems in mathematics based on using a recursive formula that generate the complete sequence of numbers. see [1] But before getting into that we begin with some Ramanujan notable…

We prove a kind of bilateral semi-terminating series related to Ramanujan-like series for negative powers of $\pi$, and conjecture a type of supercongruences associated to them. We support this conjecture by checking all the cases for many…

In this article we give the theoretical background for generating Ramanujan type $1/\pi^{2\nu}$ formulas. As applications of our method we give a general construction of $1/\pi^4$ series and examples of $1/\pi^6$ series. We also study the…

For an arithmetical function $f$, its Ramanujan expansion is a series expansion in the form $f(n)=\sum\limits_{k=1}^{\infty}a(k) c_k(n)$ where $a(k)$ are complex numbers and $c_k(n):= \sum\limits_{\substack{m=1\\(m, k)=1}}^{k}e^{\frac{2\pi…

We show how to find series expansions for $\pi$ of the form $\pi=\sum_{n=0}^\infty {S(n)}\big/{\binom{mn}{pn}a^n}$, where S(n) is some polynomial in $n$ (depending on $m,p,a$). We prove that there exist such expansions for $m=8k$, $p=4k$,…