SUMMATION OF TYPES OF SERIES 1ST 



A PHYSICAL REPRESENTATION OF THE SUMMATION 

 OF CERTAIN TYPES OF SERIES. 



L. P. SIEG. 



Most of us are on the lookout for concrete illustrations of 

 abstract ideas. To be able to visualize a mathematical process 



is to many of us a step toward the better understanding of that 

 process. The following brief discussion, although having no 

 pretentions to absolute originality, is offered as a physical il- 

 lustration of the summation of certain simple geometric series. 



In the accompanying four diagrams of figure 17 we have 

 a series of four sketches of combinations of simple machines. 

 In each case a weightless platform supporting a man of weight 

 W is suspended from the point a of the weightless, frictionless 

 lever Fab. The fulcrum of this lever is at F. The point of ap- 

 plication of the force /' which the man exerts, in the manner 

 shown by the arrow in each case, is at the point b. The force 

 /' is transmitted to the point b by the frictionless fixed pulleys 

 1\ and P 2 in diagrams 1 and 3, and by the frictionless fixed 

 pulley P ± in diagrams 2 and i. The mechanical advantage of 

 the lever is considered to be m, in order to make the problem 

 general, where m is greater than unity in diagrams 1 and 2, 

 and less than unity in diagrams 3 and 4. 



The problem in each case is to determine the force / that will 

 place the system in equilibrium. This is of course a very simple 

 physical problem. However, there are at least two ways of ap- 

 proaching the solution of the problems, and it is in the results 

 from these two methods of approach that we find the ideas 

 involved in this paper. 



Consider diagram 1. We can solve this problem algebraically 

 by equating the weight of the man plus the reaction or the 

 force /, which he exerts, to the upward force / multiplied by /". 

 the mechanical advantage of the machine. Fab. This gives us 



W + f =in f (1) 



or /=/!'/ Cm- — I) (2) 



The second manner of attacking the problem is in an ap- 

 proach by an infinite series. The man can be considered as in 

 readiness to exert the proper force, and he indulges in the fol- 

 lowing reasoning. First he knows that if his weighl is IV he 

 must pull with a force of W/m in order to support himself. 

 But this force will create an additional thrust mi the platform 



