88 EXPANSION OF FUNCTIONS. 



115. Let f(x)=a x . 



By Arts. 95 and 79, we have 



a; 2 x n 



a" = 1+ x log a + . (log a) 2 + ... + . (log a) 

 LfL I n 



Hence, changing a to e, and remembering that log e=l, 

 x* x* sc n o; n+ V* 



x n+1 e 0x 

 The term -j - - may be made as small as we please by 



sufficiently increasing n. Hence we obtain an infinite series 

 for e x , namely, 



Put x\, and we have 



"= 1 + 1 + g + ]I + (T + - 



This series may be used for calculating the approximate 

 value of e, and we may shew from it that e must be an in- 

 commensurable number. See Plane Trigonometry, Chap. x. 



It is found that e = 2'718281828.... 



116. Let f( x } 



By Arts. 95 and 78, 



x s x s 



Similarly cos x=l , H r; * 



11 li 



M fmr\ . *> . n /"^"^ i_x4i.l 



, COS f ]-fi ^T COS[ gT- TT+ftzy. 



