140 Algebra. 



distance remaining between it and the point B, so that the dis- 

 tance from A to the moving point can never equal AB, but as the 

 moving point can be brought as near as we please to B, its dis- 

 tance from A can be made to differ from the distance AB by an 

 amount as small as we please. 



Thus we see that the distance from A to the moving point ful- 

 fills all the requirements of the definition of a variable, and the 

 distance AB all the requirements of the definition of a limit. 



The student must note that it is not the poiiit B that is the 

 limit of the moving point, although the moving point approaches 

 the point B, but it is the distance AB that is the limit of the 

 distayice from A to the moving point. 



If we call the distance the point moves the first second i (then 

 of course the whole distance AB would be 2), the distance trav- 

 ersed the second second would be \, that traversed the third second 

 would be I, and so on, and the entire distance from A to the 

 moving point at the end of n seconds would be the sum of 71 



terms of the series 



T L i L JL 



^ > '2 » 4 ' 8 » 1 6 ' • • • 



Now it is sure that the more terms of this series that are taken 

 the less does the sum differ from 2 ; but the sum can never equal 

 2. Hence we say that the limit oi the sum of the series 



i+i+i+i+1^6 • • • 

 as the number of terms is indefinitely increased is 2. 



Again consider any regular polygon inscribed in a circle, and 

 then join the vertices with the middle points of the arcs subtend- 

 ing the sides, thus forming another regular inscribed polygon of 

 double the number of sides. From this polygon form another of 

 double its number of sides and so on. Now the polygon is always 

 ivithin the circle, and hence the area of the polygon can never 

 equal the area of the circle, but as the process of doubling the 

 number of sides is continued, the less does the area of the poly- 

 gon differ from the area of the circle. Hence, we say that the 

 limit of the area of the polj^gon is the area of the circle. 



Again as a straight line is the shortest distance between two 

 points, any side of the inscribed polygon is less than the sub- 

 tended arc, hence the sum of all the sides or the perimeter of the 

 polygon is less than the sum of all the subtended arcs or the cir- 



