In this paper we present a simple formula that can often be used to assist in the approximation of the sum of an alternating series

The ideas presented here were motivated by translating Euler's papers [1] and [2]. Suppose we begin by summing a terms of this series exactly, then the approximation that we will obtain is given by

Tangential quadrilaterals are defined to be quadrilaterals in which a circle can be inscribed that is tangent to all four sides. It is well known and easy to prove that a convex quadrilateral is tangential if, and only if, the angle bisectors of all four vertex angles are concurrent at a point, which is the centre of the inscribed circle (incircle). The most well-known and in problem solving useful characterisation of tangential quadrilaterals is Pitot's theorem, which states that a convex quadrilateral is tangential if and only if its consecutive sides a, b, c, d satisfy the relation a + c = b + d [1, pp. 64-67]. If you want to have more background information about characterisations of tangential quadrilaterals, then we recommend you to check out the lovely papers [2, 3, 4], as well as our previous contributions on the subject [5, 6, 7]. These six papers together include about 30 characterisations that are either proved or reviewed there with references.

In an earlier issue of the Mathematical Gazette, Nick Lord established the familiar formula for the sum of the first n squares using a physical argument based on the centroid of a configuration of masses in the plane [1]. In [2] we demonstrate an alternative configuration that gives the same result. This article is a follow-up to these papers, in which we find physical derivations of the formula for

for each k ∈ {1, 2, 3, 4, 5}. Let us first summarise the required theory. Take any region X ⊆ ℝ2 with uniform density and total area A. The centroid of is the arithmetic mean position of all of the points in X. If a region has a line of symmetry then the centroid will be located on that line.

In § 1 of this Note we first investigate some geometrical inequalities connecting a set of n unordered points in a plane. An ellipse is associated with the set of points, and use is made of the invariants of this ellipse under rotation of the coordinate axes. Some inequalities, first obtained by Weitzenböck, are derived and extended. A new geometrical entity, which we call the mutual area, is then defined for a set of n and a set of m points with the same centroid. Using the mutual invariants of two ellipses with a common centre under rotations of the coordinate axes, we obtain some inequalities connecting two sets of points with a common centroid.

In the preface to the first edition (1897) of his book [1] on the theory of finite groups, William Burnside wrote: ‘The subject is one which has hitherto attracted but little attention in this country; it will afford me much satisfaction if, by means of this book, I shall succeed in arousing interest among English mathematicians in a branch of pure mathematics which becomes the more fascinating the more it is studied.’ He returned to this point in his presidential address delivered to the London Mathematical Society on 12 November 1908.