Sum of a Series of Cosecants. 113 



3. The first stage of the analysis is now complete ; it 

 remains to transform the integral, which has just been 

 obtained, into a form suitable for calculation when n is 

 large. The following method is the only one, of several 

 which I investigated, which produces the desired result : 



First transform the integrand thus, 

 l- g -(n-i)f 1 2e~ nt f 1 1") 



{l + e- nt ){e t -l) ~V-1~ l + e- nt \ et-l + 2) 



= F 1 £1*1 Ytl _ 2e ~ nt / _L_ 1 i 1 



the terms e~ l \t are inserted to secure the convergence of the 

 integrals of the expressions contained in square brackets. 

 It is well known that 



Lte-?)^ 



where y is Euler's constant (see, e. g., Whittaker and Watson, 

 ' Modern Analysis,' p. 240). 



Now* 



e'-l~* 2^ A1 2! '4! ' * v ' (2r)I 



t 2r+1 



where < <9 < 1 when £>0, r is any positive integer, and 

 B 1? B 2 , ... are Bernoulli's numbers. 



On substitution in the second part of the integrand con- 

 nected with 7rS n , we get 



/■»°° ( e -t 2p~ nt 1 



+ H i^4i/-) mB »^! +( - )r+m+l ww}^ 



dt+2 ± i e -"-^Yi + ^ 



= 2 1 



»,,-<_<?-»* 



' (-)«B„., , , ^ 1 (-)'-+iB r+l T 

 where 



I, 



Jo l+^- w ' _ ^ m+1 Jo **+l ; 



and 0<^!<1, by the first mean-value theorem, because 

 O<0;<1 and the integrand involved in I 2r+1 is positive. 

 * Bromwich, ' Infinite Series,' p. 234. 



