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CHAPTER VII. 



EXAMPLES OF EXPANSION OF FUNCTIONS. 



114. WE shall first apply the formulae of the preceding 

 Chapter to expand certain functions. 



Required the expansion of (1 + #)"*, ra not being assumed 

 to be a positive integer. 



If /(*)-(! + *)-, 



we have /' (x) = m (1 + x)" 1 ' 1 , 



/" (x) = m (m - 1) ... (m-n+ 1) (1 + )"-", 

 /" +1 (a?) = m (m- 1) ... (m - n) (1 + xf^ 1 ; 

 hence / (0) = 1, /' (0) = TO, /" (0) = m (m - 1), . . . 

 Therefore, by Art. 95, 



(ml) 2 m(m l)...(w 

 y 2 ^ ' 



If a; be less than 1 the last term can be made as small as 

 we please by sufficiently increasing n, and in that case the 

 infinite series 



m (m 1) 2 



can, by taking a sufficient number of terms, be brought as 

 near as we please to (1 + #)"*. 



