INFINITE SERIES 121 



! 



3) -j j-^j a p fli cl 02 c W 3 .... x ci+2c ** c *+ . . . 



The coefficient of x k (k an integer) in the expansion of (i) is found by taking 

 the sum of all the terms (2) or (3) for the different combinations of p, Ci,c 2 , 

 c 3 , . . . . which satisfy 



Ci + 2C 2 + 3C 3 + . = k, 



p + ci + c 2 + c 3 + . . . . = n. 

 cf. 6.361. 



In the following series the coefficients B n are Bernoulli's numbers (6.902) 

 and the coefficients , Euler's numbers (6.903). 



6.400 



i. sin x = x -, + -:- + .... 



n=o 



CO 



2. 



3 15 315 



00 



ln-\ 



3 5 7 



' (> * = *~3~45* ~945* ~4725* 



00 





6. esc x = - + , x + , x 3 + -^, x 5 + . . . 



6.41 



2-3^ 2-4-5 2-4-6-7' 



= - - cos- 



^i 



-1 



