Let g(x)= f(x) , where f(x) is a twice differentiable positive function on (0, ) such that f(x+1)=x f(x) . Then, for N=1,2,3, . g^ (N+ 1 2 )-g^ ( 1 2 ) is equal to

Step-by-step solution

Since, f(x)=e^ g(x) aligned e^ g(x+1) & =f(x+1) & =x f(x) & =x e^ g(x) aligned and g(x+1)= x+g(x) g(x+1)-g(x)= x Replacing x by x- 1 2 , we get aligned g (x+ 1 2 )-g (x- 1 2 ) & = (x- 1 2 )= (2 x-1)- 2 g^ (x+ 1 2 )-g^ (x- 1 2 ) & =- 4 (2 x-1)^2 aligned Substituting, x=1,2,3, , N in Eq. (ii) and adding, we get g^ (N+ 1 2 )-g^ ( 1 2 )=-4 1+ 1 9 + 1 25 + + 1 (2 N-1)^2 .