Saturday, October 22, 2011

Reciprocals of Binomial Coefficients




 
(1)
RBC.ID.1:   Trigonometric Identity

(2)

proof

(3)

(4)

by geometric series

 
(5)
replacing (5) by,


(6)
results


(7)
guided by the following identities:

(8)



(9)

(10)

from (7) , (8) , (9) , and (10) ,  The first part of (7):


(11)

The first part of (11) equals zero


(12)




(13)

(14)
from (7) , (8) , (9) , and (10) ,  The second  part of (7):


(15)

(16)

(17)


(18)


from (14 and (18) ,



(19)
if x=1/2 , then



(20)
RBC.ID.2:    Derivative of Trigonometric Identity-1


 (21)

Proof  

differntiate (19)  in respect to x
 
(22)

(23)





(24)




multiply and divide  the first part of right-hand side by







(25)



(26)
from (25) and (26) 

(27)





(28)


if x=1/2 





(29)

divide both sides by 4

(30)



RBC.ID.3:    Derivative of Trigonometric Identity-2




Proof


(31)

(32)
 

(33)

Using geometric mean

(34)


(35)


Replacing





(36)





(37)


(38)

From  (31) and (38) , Therefore ,



(39)