Question

Let `g(x) = ln f(x)` where f(x) is a twice differentiable positive function on `(0, oo)` such that `f(x+1) = x f(x)`. Then for N = 1,2,3 `g'(N+1/2)- g'(1/2) =`
A. `-4{1+1/9+1/25+...+1/((2N-1)^(2))}`
B. `4{1+1/9+1/25+...+1/((2N-1)^(2))}`
C. `-4{1+1/9+1/25+...+1/((2N +1)^(2))}`
D. `4{1+1/9+1/25+...+1/((2N+1)^(2))}`

Answer 1

Correct Answer - A
Since, `f(x) = e^(g(x)) rArr e^(g(x+1))=f(x+1) = xf(x) = xe^(g(x)) `
`and" "g(x+1) = log x + g (x) `
i.e.`" " g(x+1) - g(x) = log x ` …(i)
Replacing x by ` x - 1/2 `, we get
`g(x+1/2)-g(x-1/2) = log(x-1/2) = log (2x-1) - log 2`
`:. g'(x+1/2)-g'(x-1/2) = (-4)/((2x-1)^(2))` ....(ii)
On 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/((2N-1)^(2))}`,