INVERSE METHOD OF DEFINITE INTEGRALS. 129 



Now this is the case with (1 — e')''''"^ which is also when expanded 

 of the same form as the part between the brackets; hence equating like 

 terms, we have 



Hence A,= -\.T, A,= '^^^^ .S" ^„ + , = (- 1)". (« + !)"; 



and therefore 



X„ = l-p2"^+'^^.3'7^- (-1)". (« + !)«. r. 



Cor. 1. When ?^ and n' are unequal, then ftL„'\,„ = 0. 



But when Ti'=fi, we need only take the last term of L,„ namely, (h. 1. /)"; 

 hence 



j;x„z>„ = j;(h.i.^)"{i-^.2"^+'i^j^^.3"^^-&c.| 



= (-l)..,...S....„{.-f.l.^^).l-.e.} 



_ (-l)''.1.2.3....w 

 ~ ft + l 



Cor. 2. j;x„(h.l.^)^' 



= i - ly .1 . 2 . 3....X ll'-^-' - n Q.""-' + ^~^ .3"-"-' - kc.\ 



= ( - 1)"- M .2.3 ...x A" . (A"-*-'), 



h being put = 1 after the operation of taking the «* finite diiFerenc<» 

 on the supposition that the increment of k is unity ; from whence it 

 is easy to deduce 



^^■^' = <-')-'^--^- 



Cor. 3. All the roots of the equation X„ = are real, and lie between 

 and 1. 



For if we put h. 1. (/) = it, and X„e"= U, 



then ;x„ (h. 1. ty = f. Uu^ = ti^f^ U- xw-^f.: U+ ^4^^ // U, &c. 



