Bit 



MATHEMATICS. ALGEBRA. 



[THE BINOMIAL THEOREM. 



(4). To show that 

 1 



For 



-(1 



Hence, by the general formula (IV.), thin equals 



Whence the result 

 (5). To ascertain whether the scries 1 + nx -\ -- JT2 



** + ... U convergent 



For the sake of brevity, let us write the series 



where A, - 



F+T 



+ 1 i 

 "-~ 



r 1 n+_l_, 

 r+^ "r+2 - 



Now, r can be taken so large that T 2 



< 1. So that if 



n r 



< L 



1 ; and there- 

 n r 1 



r+2 



l.s 



also < L 



Also, these fractions are all negative, . . if A, be posi- 

 tive, A,+t is negative, A,^.. is positive, and so on. But if 

 we neglect the consideration of the signs from what has 

 been proved, it appears that A^ > A, +1 , A' +1 > A r+z , 

 <tc. 



Hence, the part of the series beginning with A,ar; i.e. 

 A,af + A, + , + z"M + A r +r + s+ ---- 



has its terms (if x be positive) alternately positive and 

 negative, and is term by term less than 



A, * 



provided A, < 1. 



But this latter series equals 



x+ac 1 )(fce. 



which is convergent of z < L 



Hence, provided x < 1, the series, at all events, after 

 a certain number of terms, converges. 



(6). Hence, if x < L 



is true arithmetically. 

 formula to extractin 

 extract the 6" root of 1-L The fifth root of I'l, is 



We may therefore apply this 

 formula to extracting roots of numbers. Thus to 



Now, 



1-4-9 



1 -4 -9 -14-19 

 6 10-15-20-26- 



4 1 

 6-10- 10" 



T^T - '0008 



1-4-9 . _1 

 6-1016 10 



1-1914 



0008 

 0000336 



which is the S^ root of I'l, true to 5 places of decimals. 



J^tj.t A'x 3 



1L To Aow that a.- 1 + Ax +- + + ---- 



ad infinilum. 



A - 1_ ( - + ^ 



This is called the Exponential Theorem or series. 

 By the Binomial Theorem. 



(o !/ + .... (a) 



Now, it is plain that if we multiply the factors of the 

 coefficient* of (o 1)', (a If, (a 1) . . . . together, 

 we may re-arrange the series so that it shall become 



o-l + Ax + Bz 1 +C.r s + Dz + . . . . (5) 

 where A, B, C, D, . . . . contain (o 1) and its powers 

 in some determinate manner. For example, if wo ex- 

 amine (a), we shall find that each term, after the first, 

 contains the first power of x ; viz., the second term con- 



tains 1(0 1), the third contains *. ^ , the 



fourth contain* x. ^-q- , the fifth 



and 



1 -4 -9 -14- 

 ~ 6-10-15-20- 



= 1 



- -02 



- -000048 



1020048 

 000803,36 



. 1-019245 



so on ; hence the term involving the first power of x 

 when (a) is re-arranged, must be 



-j-K... 



In like manner, we might find B.C.D ..... in scries (5). 

 Instead of doing so, we proceed as follows : since (6) is 

 true for all values of x, we must have 



a = l + Ay + By + Cv + Dy-.... (d) 



Now, o* X o= a* + 



But, o* X o = 1 -f Ax + Bx s +Cjr 1 +Dj; 4 + . . . .} 

 + y { A -f A* + ABx + ACx 3 + ....} 



+Cx 



VJD+. 

 + '. 



