An Infinite Series for π with Determinants

Abstract

This can be proved with some geometry and calculus. A quarter of the area of a circle of unit radius can be expressed as the sum of an infinite series: 7T 4 =O+2?A,+2lA2+2 2A3+ * * 2tz-Sl where the Ai (i = 0,1, 2, ...) are the areas of the isosceles triangles indicated in FIGURE 1. Let Q and PO be the points (1,0) and (0,1), respectively, and Pt, be the apex of the triangle of area A (n?>1). Po = I P a P2