For 10 points, name these series which consist of terms r to the k summed over k, which are often contrasted with arithmetic series.
A curious one of these entities, which can be said to be equal to 1 or 0, is named for Grandi. Archimedes used one of these objects to calculate the area of the title conic section in his treatise The Quadrature of the Parabola. The St. Ives riddle can be restated as solving a finite one of these. The (*) "hyper-" form of these entities involves summing rational functions of k; that is, the quotient of polynomials P(k) and Q(k). These converge if the absolute value of the power of each term is less than 1, in which case they are equal at infinity to 1 over the quantity 1 minus the first term. For 10 points, name these series which consist of terms r to the k summed over k, which are often contrasted with arithmetic series.
A curious one of these entities, which can be said to be equal to 1 or 0, is named for Grandi.
{geometric} series