Related papers: A new elementary proof of the formula $\sum\limits…
We give a proof of the identity $\zeta(2)=\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}6$ using the fundamental theorem of calculus and differentiation under the integral sign.
The number $\frac{\pi ^{2}}{6}$ is involved in the variance of several distributions in statistics. At the same time it holds $\sum\nolimits_{k=1}^{\infty }k^{-2}= \frac{\pi ^{2}}{6}$, which solves the famous Basel problem. We first provide…
We give another proof for \[ \sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6} \] that basically follows from the theory of difference equations.
In this shortnote, a series expansion technique introduced recently by Dancs and He for generating Euler-type formulae for odd zeta values $\:\zeta{(2 k +1)}$, $\zeta{(s)}$ being the Riemann zeta function and $k$ a positive integer, is…
A simple proof of Euler's formula which states that the sum of the reciprocals of all natural numbers squared equals $\pi^2/6$ is presented based on the distribution theory introduced by Laurent Schwartz. Additional identities are obtained…
Using a double integral we give another solution to the Basel Problem
This short note is a comment on a historical aspect of a famous formula dating from the 18th century.
We present an astonishingly simple and elegant proof of the celebrated Basel problem.
By doing a slight change to a beautiful and widely unknown argument by E. L. Stark [E. L. Stark, Application of a Mean Value Theorem for Integrals to Series Summation, Amer. Math. Monthly 85 (1978) 481--483.] we get a candidate to be…
A new simple geometric method is presented for finding the exact value of $\sum_{n=1}^\infty 1/n^2$.
Let $\{q_n\}_{n=0}^\infty\subset [0,1]$ satisfy $q_0=0$, $\sum_{n=0}^\infty q_n=1$, and $\gcd\{n\geq 1\mid q_n\neq 0\}=1$. We consider the following process: Let $x$ be a real number. We first set $x=0$. Then $x$ is increased by $i$ with…
The Basel problem, solved by Leonhard Euler in 1734, asks to resolve $\zeta(2)$, the sum of the reciprocals of the squares of the natural numbers, i.e. the sum of the infinite series: \begin{equation}…
The Basel problem consists in finding the sum of the reciprocals of the squares of the positive integers. It was finally solved in 1735 by Leonhard Euler. In this paper, we propose a simple proof based on the Weierstrass Sine product…
Euler's solution in 1734 of the Basel problem, which asks for a closed form expression for the sum of the reciprocals of all perfect squares, is one of the most celebrated results of mathematical analysis. In the modern era, numerous proofs…
The purpose of this paper is to present series expansions for even powers of the number $\pi$. This is accomplished by generalizing Euler's method for solving the Basel Problem, which was published in 1735. We employ elementary symmetric…
We obtained a new formula for $\pi$.
Because of its relation to the distribution of prime numbers, the Riemann zeta function {\zeta} (s) is one of the most important functions in mathematics. The zeta function is defined by the following formula for any complex number s with…
We establish a connection between a function and a series representation using a similar technique with that that Euler used to solve the Basel problem. Our result concerns a more general series from which one can obtain $\zeta(2k)$ as a…
In 1859, Riemann had announced the following conjecture : the nontrivial roots (zeros) $s=\alpha+i\beta$ of the zeta function, defined by: $$\zeta(s) =\displaystyle \sum_{n=1}^{+\infty}\frac{1}{n^s},\,\mbox{for}\quad \Re(s)>1$$ have real…
Discretizations of differential equations are often studied through their modified equation. This is a differential equation, usually obtained as a power series, with solutions that exactly interpolate the discretization. By comparing the…