SUMMATION OF TYPES OF SERIES 189 



l/m + l/m 2 + l/m r +l/m 4 + = l/(m—l) (4) 



This is the correct summation of the series and the series is 

 convergent, since we assumed m to be greater than unity. Eence 

 the two methods of approach are equally good, and both lead to 

 the correct answer. 



It is a matter of some interesl to speculate as to which method 



would be used by a man on an actual platform of this kind. 

 It seems that the algebraic method would certainly not be used. 

 Either his muscles would gradually exert tension in the manner 

 represented by equation (3), or else he would approach the 

 correct force by an oscillatory muscular pull, the oscillations 

 gradually getting smaller and smaller until the correel force 

 / has been reached. This type of series will be found in the 

 discussion below. Such a problem as this, aside from these psy- 

 chological aspects, cannot help but be of some value to a teacher 

 of elementary physics or mathematics in that it gives a tangible 

 meaning to an infinite series. Of course there are many other 

 problems that will illustrate this particular point. 



Consider now the arrangement shown in diagram 2. The re- 

 action is now opposite in direction to W, and the algebraic so- 

 lution is given by 



W—f=mf (5) 



or f=W/(m+l) (6) 



The other method of obtaining f is somewhat similar to the 

 preceding one. A first pull of W/m is necessary. This pull, 

 however, decreases the load by W/m, and therefore the tension 

 in the rope must be slacked by an amount W/m 2 . This in turn 

 adds to the thrust on the platform of W/m 2 , and an additional 

 pull of W/m 3 must be exerted. In short the force is determined 

 by the following series: 



f=W/m—W m i +W/m 3 —W/m i + (7) 



Equating equations (6), and (7), and cancelling the W's we 

 obtain 



1/m— l/m*+l/m*— l/m 4 +. . ..=l/(m+l) (8) 



As long as m is greater than unity this is a convergent sei 

 and is correctly summed. Here again then we have the two 

 methods of attack leading in one case to a simple answer, and 

 in the other to an infinite converging series, the series being 

 a correct representation of the algebraic result. 



