16 N. F. DUPUIS : 



The developmeuts of tlie series wMch are the equivalents of a', siu 0^ taù ~^ x ac- 

 cording to ascending x^owers of the variables are among the most important developments 

 in the vvfhole of mathematics, and usually occur quite early in a student's course. These 

 developments are generally effected through the differential calculus, or by some means 

 employing the method of limits. The latter method involves principles which always 

 seem questionable to a beginner and to which he becomes reconciled only after much 

 thought and many applications. The method here presented is inductive and is free 

 from the seemingly questionable feature mentioned. 



It is necessary in what follows to premise the following : — 



The symbol C" will denote the number of combinations of n things when taken r 

 together ; then from the relation existing between combinations and the binomial series 

 we have : — 



Cï + C; + C;; + 4C;; ?* even "I 



" : [ = 2"-'-l (1) 



C;;_i 71 odd J 



Cî"+Cr+C5"+ C^i neven-) 



f=2^--=' (2) 



iCf n odd 3 



a:"+C|" + CÎ»+ Wf neven") 



[=2-=-l (8). 



C;,^.! n odd ) 



II. 



Development of the Exponential Series without the Use of Limits. 



To develop a', assuming that the development can be expressed in ascending powers 

 of a;. 

 Assume 



a" = l-\-a^x-\-a.,x-+a.^x"+ .... a„x + . . . . 



The first term of this assumption is correct ; since when z becomes zero, a'^ becomes unity. 



Taking the property or' = («')", 

 and substituting we obtain. 



and 



a'=^- = l+2aia;+2'«ja''+ 



. 2"aX+ • • 



(a^)-= l + 2ap:-\-2a.. 



x'+ 



2a" 

 2a„_,a, 



a„a„ 



2 2 



0,-"+ . 



n even ; 



n odd. 



