Pi Formulas: some smooth stones on the beach of rough numbers

This article is about Pi Formulas, infinite series of fractions which sum to multiples of Pi. Each such one can be associated with a unique set Sₖ of rough numbers, where k is a prime number. Given Sₖ for any prime k, the set S_k^, where k^ is the smallest prime greater than k, can be constructed easily. From this it follows that Pi Formulas occur in disjoint families. In any family, there is a first member, a series of least prime number kₘᵢₙ, which must be summed from first principles. Then from the series of some k > kₘᵢₙ already summed, the series for k^ can be summed by a simple algebraic procedure. A good number of Pi Formulas, belonging to a variety of families and giving results believed to be new, are presented here.