INFINITE SERIES 1 17 



6.31 Convergence of the binomial series. 



The series converges absolutely for | x \ <i and diverges for | x \ >i. 

 When x = i, the series converges for n> -i and diverges for n^ -i. It is abso- 

 lutely convergent only for n>o. 



When x = -i it is absolutely convergent for n>o, and divergent for n<o. 



6.32 Special cases of the binomial series.- 



(a + b) n = a n ( i + -) = b n 



If 



<i put x = - in 6.30; if 



a . 



> i put x = - in 6.30. 



6.33 



, ,^ _ w n(m n) 2 n(w - n) (2W - n) 3 



m 



2m 



,_ x* n(m - n) (2m - n) ..... \_(k - i)m - n] k 



V *) 7~~t 1, % 



klm k 



2. (i + X)~ l = I - X + X 2 - X 3 -f X 4 - 



3. (i + x)~ 2 = i - 2X + 3^ 2 - 4^ 3 + 5 



2 2-4 



-4- 



-4-6- 



x , 113 o H3V5 .3 , i-3'5-7 X 4 



1 3/-T A - A, -\- /-o**' -. 



2 2-4 2-4-6 2-4-6-8 



3' 6> 9 3-6-9- 





9. (i + *)-i = 



i - a? -f x - 



2 2-4 2-4-6 



-4- 2-4- 



x* + 



-4- 



10. 



ii. 



12. 



4 32 



-a: + - 

 4 



128 2048 



32 



-a; 

 25 



128 



x 

 125 



2048 



-^- x* 

 625 



