The QEC Sum (Johnson Sum) of the Unit Fractions

In Quaternionic Eulerian Calculus (QEC) a tangent is a derivative and the area it encloses is an integral. On this basis a summation is defined, called the QEC sum or Johnson sum, that takes the derivatives (tangents) at every point of a set on a curve and returns the integral they share. For the curve y = 1/x, every unit fraction 1/n is a point whose tangent cuts from the axes a right triangle of area exactly 2, and a squeeze symmetry of the hyperbola carries each such triangle onto every other. The QEC sum of all the unit fractions, 1/1 included, is therefore 2. The QEC sum is a named summation, in the same standing as the Cesàro, Abel and Ramanujan sums; it is distinct from term-by-term addition, under which the unit fractions have no finite total. The same invariant appears in circuit theory: on the resistance–conductance hyperbola and in the maximum power transfer theorem.