In this article, I suggest that calculus instruction should include a wider variety of examples of convergent and divergent series than is usually demonstrated. In particular, a number of convergent series, such as ∑ k≥1 k3 2k , are considered, and their exact values are found in a straightforward manner. We explore and utilize a number of mathematical topics, including manipulation of certain ...

The aim of the paper is to relate computational and arithmetic questions about Euler’s constant γ with properties of the values of the q-logarithm function, with natural choice of q. By these means, we generalize a classical formula for γ due to Ramanujan, together with Vacca’s and Gosper’s series for γ, as well as deduce irrationality criteria and tests and new asymptotic formulas for computin...

In this paper we investigate the two-dimensional compressible isentropic Euler equations for Chaplygin gases. Under the assumption that the initial data is close to a constant state and the vorticity of the initial velocity vanishes, we prove the global existence of the smooth solution to the Cauchy problem for two-dimensional flow of Chaplygin gases.

Let m be a positive integer, and pn(m) the proportion of permutations of the symmetric group Sn that admit an m-th root. Calculating the exponential generating function of these permutations, we show the following asymptotic formula pn(m) ∼ n→+∞ πm n1−φ(m)/m , where φ is the Euler function and πm an explicit constant.

We show that sequences of compact gradient Ricci solitons converge to complete orbifold gradient solitons, assuming constraints on volume, the L-norm of curvature, and the auxiliary constant C1. The strongest results are in dimension 4, where L 2 curvature bounds are equivalent to upper bounds on the Euler number. We obtain necessary and sufficient conditions for limits to be compact.

Given a compact four dimensional manifold, we prove existence of conformal metrics with constant Q-curvature under generic assumptions. The problem amounts to solving a fourth-order nonlinear elliptic equation with variational structure. Since the corresponding Euler functional is in general unbounded from above and from below, we employ topological methods and minimax schemes, jointly with the...

Abstract We consider the interface problem between two incompressible and inviscid fluids with constant densities in the presence of surface tension. Following the geometric approach of [14, 15] we show that solutions to this problem converge to solutions of the free–boundary Euler equations in vacuum as one of the densities goes to zero.