Sum of a Series of Cosecants. 



117 



n. 



s». 



n. 



B n . 



2* 



1-00000 



25 



5322602 



3* 



2-30940 



26 



56-00452 



4* 



3-82843 



27 



58-80752 



5* 



5-50552 



28 



61-63406 



6 



7-30938 



29 



64-48341 



7 



9-21906 



30 



67-35460 



8 



11-21946 



35 



82-01599 



9 



13-29332 



40 



97-13358 



10 



15-44980 



45 



112-65005 



11 



17-66388 



50 



128-52084 



12 



19-93579 



55 



144-71045 



13 



22-26070 



60 



161-18983 



14 



24-63456 



65 



177-93472 



15 



27-05387 



70 



194-92471 



16 



29-51559 



75 



212-14226 



17 



32-01709 



80 



229-57213 



18 



34-55603 



85 



247-20108 



19 



37-13041 



90 



265-01741 



20 



39-74028 



95 



283-01061 



21 



42-37770 



100 



301-17141 



22 



45-04863 



360 



1377-78855 



23 



47-74646 



1000 



4477-59415 



24 



50-47299 







* The values of 8 n for n=2, 3, 4, 5'^were calculated directly and not from 

 the asymptotic expansion. 



By the methods employed in §§ 2, 3 we get 



t o C J n &\ (l-e-^-^dt 

 7rT „=2.J o cosh^). A-_-^ 



= 2^ ooBh(V»0- (1 + g - <rf) y_ 1) » 

 where \ = /3/(2tt) ; it then follows that 



A =2ra £{ C °;y-^}^ + 2n[ 



-t 0-nt 



+ 2n 



2e~ ni cosh (\nt) 



{? 



t J 



1 {Knt) \ 



n+e-^yr j 



dt 



dt 



+ 4 S 



(-)-B, 



where 

 and 



-I 



du 



m to cosh Xw 



(_y + i Br+l T 



e u -\-l 

 O<0 S <1. 



