Money, Bank Credit, and Economic Cycles - The Ludwig von Mises ...

214 Money, Bank Credit, and Economic CyclesThe above expression represents the sum of the terms in ageometrical progression. The terms increase and have a commonratio of 0.9. 27In our example, r=0.9 and a=1,000,000 m.u., and hence thesum of the terms would be equal to:[13] a = 1,000,000 = 1,000,000 = 10,000,000 m.u.1 – r 1 – 0.9 0.127 The sum of the sequence:[9] Sn = a + ar + ar 2 ... + ar n-1 ; if multiplied by the common ratio r,is:[10] rSn = ar + ar 2 + ar 3 ... + ar n-1 + ar n ; by subtracting [10] from [9],we obtain:Sn – rSn = a – ar n ; and factoring out the common factor on bothsides:Sn(1 – r) = a(1 – r n ); then we isolate Sn:[11] Sn = a(1 – rn ); and when r < 1, r n approaches 01 – ra(1 – r n ) aand the Lim Sn = Lim=n Z ∞ n Z ∞ 1 – r 1 – r ; if |r| < 1.Therefore we may conclude that:[12] Sn =a1 – r; if |r| < 1The Greek sophist Zeno was the first to pose the problem of adding theterms in a sequence with a common ratio less than one. He addressedthe problem in the fifth century B.C., posing the well-known question ofwhether or not the athlete Achilles would be able to catch the turtle. Theproblem was not satisfactorily solved, however, because Zeno failed torealize that infinite series with a common ratio less than one have a convergentsum (not a divergent sum, like he believed). See The ConciseEncyclopedia of Mathematics, W. Gellert, H. Kustner, M. Hellwich and H.Kastner, eds. (New York: Van Nostrand, 1975), p. 388.