历史与综述

This is a translation of an article presented by Leonhard Euler on 18 March 1776 (Opera Omnia I-XVIII, pp. 265-290) and of summaries for it by Sim\'eon Denis Poisson in 1820 and by Heinrich Burkhardt in 1916. An appendix lists in modern…

This paper does exactly what the title says it does. It expands the given series to arrive at the familiar "pentagonal number" expansion, also known as the pentagonal number theorem, and recalls its application to partition numbers. The…

This paper is about the product z^q/(1 - z)^(q + 1)(1 + (z/(1 - z)))^p, Euler gives the Talylor-Series and takes a closer look at the coefficient.

Euler evaluates the integrals in the title and recognizes a recursion between them, which he then uses to give continued fractions for the log and arctan. The paper is translated from Euler's Latin original into German.

Euler gives a continued fraction representation of (1 + x)n. involving 1,3,5,7,... and n^2-1,n^2-4,n^3-9,... and squares of z, for x=2y and y=z/(1-z). He evaluates this continued fraction at z=t sqrt(-1), for "vanishing" n, and for infinite…

This paper has more formal series with Bernoulli numbers and the Riemann zeta function. He gives the generating function for the Bernoulli numbers. The paper is translated from the Latin original into German.

Euler investigates the Taylorseries of (1+x+xx)^n and uses the results to evaluates some integrals which are today often proved with the calculus of residues.

Euler gives an asymptotic approximation for the function f(x) and recognizes that he is trying to interpolate the factorial function introduced in E19 "De progressionibus transcendentibus seu quarum termini generales algebraice dari…

This paper, along with E592 and E636, seems to consider the binomial expansion (1+z)^n in the case where z is complex. Euler even gives the sums of divergent series. The paper is translated from Euler's Latin original into German.

Here Euler notes the recursive relation for the general binomial coefficients, by assuming that (1+x)^a can be expanded in a power series.

In this paper, Euler transforms the divergent series in the title, and thereby dervies the well known continued fraction expansion for pi/4 from Leibniz's series. The paper is translated from Euler's Latin originial into German.

The title says what is done here. Euler finds a news series for the arc of an ellipse. The paper is translated from the Latin original into German.

This article presents an overview, and recent history, of studies of gender gaps in the mathematically-intensive sciences. Included are several statistics about gender differences in science, and about public resources aimed at addressing…

Euler starts with a hypergeometric series F(a, b, c, x), and differentiates it to get a functional relation. This relation is today known as Euler's identity. Then he integrates to get another and ends up with something like Legendre…

Euler defines a function f(x) somehow as an infinite product and a generalization of [x], where [x] ist, what we now call following Legendre the Gamma-Funktion. He gets some recursive relationships for f(x), by applying some very nice…

This paper has been withdrawn by the author.

This paper has been withdrawn by the author.

In this article, we suggest the use of Mathematical Relationships as a possible way to decrease the students' disinterest in Mathematics. First we made a consideration of the environment of the classroom from the perspective of teachers and…

The earliest origins of mathematics in the Indian subcontinent is generally dated around 800-500 BCE when the {\em Sulbasutras} are thought to have been written. In this article we suggest that mathematical thinking in South Asia, in…

Boris Venkov passed away on November 10 2011 just 5 days before his 77th birthday. This article gives a short survey of the mathematical work of Boris Venkov in this direction.